| L(s) = 1 | + 9·2-s + 108·3-s − 49·4-s + 972·6-s + 1.18e3·7-s − 387·8-s + 7.29e3·9-s + 5.37e3·11-s − 5.29e3·12-s + 8.42e3·13-s + 1.06e4·14-s + 3.48e3·16-s + 4.89e3·17-s + 6.56e4·18-s + 1.52e4·19-s + 1.28e5·21-s + 4.83e4·22-s + 1.10e5·23-s − 4.17e4·24-s + 7.58e4·26-s + 3.93e5·27-s − 5.82e4·28-s − 1.20e5·29-s + 1.16e5·31-s + 8.46e4·32-s + 5.80e5·33-s + 4.40e4·34-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 2.30·3-s − 0.382·4-s + 1.83·6-s + 1.30·7-s − 0.267·8-s + 10/3·9-s + 1.21·11-s − 0.884·12-s + 1.06·13-s + 1.04·14-s + 0.212·16-s + 0.241·17-s + 2.65·18-s + 0.509·19-s + 3.02·21-s + 0.968·22-s + 1.88·23-s − 0.617·24-s + 0.845·26-s + 3.84·27-s − 0.501·28-s − 0.913·29-s + 0.704·31-s + 0.456·32-s + 2.81·33-s + 0.192·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(46.36255507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(46.36255507\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 9 T + 65 p T^{2} - 153 p^{3} T^{3} + 651 p^{4} T^{4} - 153 p^{10} T^{5} + 65 p^{15} T^{6} - 9 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 1188 T + 219424 p T^{2} - 89985492 T^{3} + 253878640350 T^{4} - 89985492 p^{7} T^{5} + 219424 p^{15} T^{6} - 1188 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 5376 T + 68059592 T^{2} - 265739821920 T^{3} + 1856153151437406 T^{4} - 265739821920 p^{7} T^{5} + 68059592 p^{14} T^{6} - 5376 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 648 p T + 127784896 T^{2} - 910091284632 T^{3} + 8912329711731150 T^{4} - 910091284632 p^{7} T^{5} + 127784896 p^{14} T^{6} - 648 p^{22} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 288 p T + 990101248 T^{2} - 13265399718720 T^{3} + 453457643591452926 T^{4} - 13265399718720 p^{7} T^{5} + 990101248 p^{14} T^{6} - 288 p^{22} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 15232 T + 392774572 T^{2} - 22311843627904 T^{3} + 1443987068893239574 T^{4} - 22311843627904 p^{7} T^{5} + 392774572 p^{14} T^{6} - 15232 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 110016 T + 14943940732 T^{2} - 1034411925986496 T^{3} + 79201784745547797990 T^{4} - 1034411925986496 p^{7} T^{5} + 14943940732 p^{14} T^{6} - 110016 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 120036 T + 67036689080 T^{2} + 5675627737126332 T^{3} + \)\(17\!\cdots\!78\)\( T^{4} + 5675627737126332 p^{7} T^{5} + 67036689080 p^{14} T^{6} + 120036 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 116864 T + 81093883468 T^{2} - 6882918957523712 T^{3} + \)\(30\!\cdots\!54\)\( T^{4} - 6882918957523712 p^{7} T^{5} + 81093883468 p^{14} T^{6} - 116864 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 663768 T + 359089738528 T^{2} - 136020997240132392 T^{3} + \)\(48\!\cdots\!70\)\( T^{4} - 136020997240132392 p^{7} T^{5} + 359089738528 p^{14} T^{6} - 663768 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 253824 T + 334342452188 T^{2} - 27270984718092672 T^{3} + \)\(66\!\cdots\!34\)\( T^{4} - 27270984718092672 p^{7} T^{5} + 334342452188 p^{14} T^{6} - 253824 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1092960 T + 1046348379820 T^{2} - 610415406190516320 T^{3} + \)\(36\!\cdots\!98\)\( T^{4} - 610415406190516320 p^{7} T^{5} + 1046348379820 p^{14} T^{6} - 1092960 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 132840 T + 921587080780 T^{2} + 106414099405575480 T^{3} + \)\(48\!\cdots\!38\)\( T^{4} + 106414099405575480 p^{7} T^{5} + 921587080780 p^{14} T^{6} - 132840 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1994616 T + 4981442415712 T^{2} - 6519241402601783976 T^{3} + \)\(90\!\cdots\!50\)\( T^{4} - 6519241402601783976 p^{7} T^{5} + 4981442415712 p^{14} T^{6} - 1994616 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 545712 T + 6864766660712 T^{2} + 4026585148405309584 T^{3} + \)\(21\!\cdots\!34\)\( T^{4} + 4026585148405309584 p^{7} T^{5} + 6864766660712 p^{14} T^{6} + 545712 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 3216760 T + 11354186987116 T^{2} + 29259400663039022440 T^{3} + \)\(51\!\cdots\!46\)\( T^{4} + 29259400663039022440 p^{7} T^{5} + 11354186987116 p^{14} T^{6} + 3216760 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 2013336 T + 14331869916028 T^{2} - 24681696313094184600 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} - 24681696313094184600 p^{7} T^{5} + 14331869916028 p^{14} T^{6} - 2013336 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 690912 T + 24146587182668 T^{2} + 3374261752345932384 T^{3} + \)\(28\!\cdots\!70\)\( T^{4} + 3374261752345932384 p^{7} T^{5} + 24146587182668 p^{14} T^{6} + 690912 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 5498064 T + 40857210451732 T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(61\!\cdots\!90\)\( T^{4} + \)\(13\!\cdots\!64\)\( p^{7} T^{5} + 40857210451732 p^{14} T^{6} + 5498064 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 7190080 T + 89811290861836 T^{2} - \)\(40\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!86\)\( T^{4} - \)\(40\!\cdots\!60\)\( p^{7} T^{5} + 89811290861836 p^{14} T^{6} - 7190080 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 13158432 T + 171401132585740 T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(79\!\cdots\!46\)\( T^{4} + \)\(11\!\cdots\!08\)\( p^{7} T^{5} + 171401132585740 p^{14} T^{6} + 13158432 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 22889448 T + 297109023037052 T^{2} + \)\(26\!\cdots\!16\)\( T^{3} + \)\(19\!\cdots\!34\)\( T^{4} + \)\(26\!\cdots\!16\)\( p^{7} T^{5} + 297109023037052 p^{14} T^{6} + 22889448 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 17873136 T + 441440121485188 T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(58\!\cdots\!86\)\( T^{4} - \)\(46\!\cdots\!20\)\( p^{7} T^{5} + 441440121485188 p^{14} T^{6} - 17873136 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.374206299963391291846927504089, −8.784418888474675965628471840629, −8.742259448668033492307464474741, −8.446569554205313574028316723086, −8.020511637206834625388414480945, −7.924635135958612749838947482332, −7.41829928262291887920110907561, −7.16771377876001103157838045454, −7.04678436717967394993162538530, −6.41225730896992552558112744786, −5.96833085965788158847664305618, −5.62131824143642158155303827113, −5.35790075949097753706953542766, −4.57567679492180502566003120050, −4.30114139640549975079502041010, −4.26288130300957430163256384767, −4.17457456367466495301318606490, −3.22893205406275890583223873157, −3.06864374172009820563395935715, −3.00116470512469888020581603410, −2.10417645883994214529100837019, −1.92877385055314760706352215132, −1.23122699014621577885316607690, −1.03805817675042484863550453259, −0.77249770179707580692586814446,
0.77249770179707580692586814446, 1.03805817675042484863550453259, 1.23122699014621577885316607690, 1.92877385055314760706352215132, 2.10417645883994214529100837019, 3.00116470512469888020581603410, 3.06864374172009820563395935715, 3.22893205406275890583223873157, 4.17457456367466495301318606490, 4.26288130300957430163256384767, 4.30114139640549975079502041010, 4.57567679492180502566003120050, 5.35790075949097753706953542766, 5.62131824143642158155303827113, 5.96833085965788158847664305618, 6.41225730896992552558112744786, 7.04678436717967394993162538530, 7.16771377876001103157838045454, 7.41829928262291887920110907561, 7.924635135958612749838947482332, 8.020511637206834625388414480945, 8.446569554205313574028316723086, 8.742259448668033492307464474741, 8.784418888474675965628471840629, 9.374206299963391291846927504089
