| L(s) = 1 | − 8·2-s − 4·3-s + 40·4-s + 32·6-s − 26·7-s − 160·8-s − 14·9-s + 93·11-s − 160·12-s − 32·13-s + 208·14-s + 560·16-s − 108·17-s + 112·18-s + 185·19-s + 104·21-s − 744·22-s − 92·23-s + 640·24-s + 256·26-s + 27·27-s − 1.04e3·28-s + 294·29-s − 211·31-s − 1.79e3·32-s − 372·33-s + 864·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.769·3-s + 5·4-s + 2.17·6-s − 1.40·7-s − 7.07·8-s − 0.518·9-s + 2.54·11-s − 3.84·12-s − 0.682·13-s + 3.97·14-s + 35/4·16-s − 1.54·17-s + 1.46·18-s + 2.23·19-s + 1.08·21-s − 7.21·22-s − 0.834·23-s + 5.44·24-s + 1.93·26-s + 0.192·27-s − 7.01·28-s + 1.88·29-s − 1.22·31-s − 9.89·32-s − 1.96·33-s + 4.35·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6993207147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6993207147\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 4 T + 10 p T^{2} + 149 T^{3} + 1310 T^{4} + 149 p^{3} T^{5} + 10 p^{7} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 26 T + 724 T^{2} + 10099 T^{3} + 260534 T^{4} + 10099 p^{3} T^{5} + 724 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 93 T + 6725 T^{2} - 31627 p T^{3} + 14317128 T^{4} - 31627 p^{4} T^{5} + 6725 p^{6} T^{6} - 93 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 32 T + 976 T^{2} - 52631 T^{3} - 4682512 T^{4} - 52631 p^{3} T^{5} + 976 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 108 T + 18260 T^{2} + 93099 p T^{3} + 131243742 T^{4} + 93099 p^{4} T^{5} + 18260 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 185 T + 33223 T^{2} - 186623 p T^{3} + 363066172 T^{4} - 186623 p^{4} T^{5} + 33223 p^{6} T^{6} - 185 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 294 T + 42335 T^{2} + 2109052 T^{3} - 859672512 T^{4} + 2109052 p^{3} T^{5} + 42335 p^{6} T^{6} - 294 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 211 T + 103384 T^{2} + 13064236 T^{3} + 4073558761 T^{4} + 13064236 p^{3} T^{5} + 103384 p^{6} T^{6} + 211 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 5 T + 134830 T^{2} - 4948045 T^{3} + 8513448058 T^{4} - 4948045 p^{3} T^{5} + 134830 p^{6} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 9 p T + 291554 T^{2} + 74201826 T^{3} + 30573978075 T^{4} + 74201826 p^{3} T^{5} + 291554 p^{6} T^{6} + 9 p^{10} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 100 T + 82204 T^{2} - 9571140 T^{3} + 522019158 T^{4} - 9571140 p^{3} T^{5} + 82204 p^{6} T^{6} - 100 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 363 T + 163502 T^{2} + 6031301 T^{3} + 2659565058 T^{4} + 6031301 p^{3} T^{5} + 163502 p^{6} T^{6} - 363 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 21 T + 269018 T^{2} - 89970593 T^{3} + 27668065578 T^{4} - 89970593 p^{3} T^{5} + 269018 p^{6} T^{6} + 21 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 33 T + 343586 T^{2} + 84316061 T^{3} + 56145878106 T^{4} + 84316061 p^{3} T^{5} + 343586 p^{6} T^{6} + 33 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 307 T + 487855 T^{2} + 298124379 T^{3} + 113148939756 T^{4} + 298124379 p^{3} T^{5} + 487855 p^{6} T^{6} + 307 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 725 T + 912826 T^{2} + 351937325 T^{3} + 318440704618 T^{4} + 351937325 p^{3} T^{5} + 912826 p^{6} T^{6} + 725 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 1257 T + 1758566 T^{2} + 1300005098 T^{3} + 998915687661 T^{4} + 1300005098 p^{3} T^{5} + 1758566 p^{6} T^{6} + 1257 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 509 T + 697972 T^{2} + 203398343 T^{3} + 321017582158 T^{4} + 203398343 p^{3} T^{5} + 697972 p^{6} T^{6} + 509 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 1202 T + 1003684 T^{2} - 146546698 T^{3} + 4581734390 T^{4} - 146546698 p^{3} T^{5} + 1003684 p^{6} T^{6} - 1202 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1377 T + 1324064 T^{2} - 677164645 T^{3} + 435520077246 T^{4} - 677164645 p^{3} T^{5} + 1324064 p^{6} T^{6} - 1377 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 984 T + 2444900 T^{2} + 1775501064 T^{3} + 2545577861478 T^{4} + 1775501064 p^{3} T^{5} + 2444900 p^{6} T^{6} + 984 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 137 T + 867295 T^{2} + 1257176833 T^{3} + 651849046340 T^{4} + 1257176833 p^{3} T^{5} + 867295 p^{6} T^{6} + 137 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.55360044949997774916667933792, −6.52087869262768544828296409576, −6.36213954901504590093555987082, −6.27683672093954279489346229506, −5.99235413849677190310957446535, −5.56439446468880103450287802375, −5.41536720439755899801802541230, −5.24890504753722217841713238811, −5.01603688523649070095620499664, −4.35512192545127323722596941636, −4.25140047878725334557975372134, −3.93075915307069635047939476288, −3.91829361598142905208784365588, −3.20767236746668249780698959934, −3.04449465888307973006624247418, −2.99581138458768258465683144925, −2.81501564075239309712231906716, −2.31423921306607636359279718594, −1.79698649354786693364304529792, −1.71650897129404870767185916407, −1.44180770022479117562019226101, −1.19480429254777416439412253095, −0.49313549747766789060037310329, −0.45055148054038215001062249590, −0.41254146429255515699000915520,
0.41254146429255515699000915520, 0.45055148054038215001062249590, 0.49313549747766789060037310329, 1.19480429254777416439412253095, 1.44180770022479117562019226101, 1.71650897129404870767185916407, 1.79698649354786693364304529792, 2.31423921306607636359279718594, 2.81501564075239309712231906716, 2.99581138458768258465683144925, 3.04449465888307973006624247418, 3.20767236746668249780698959934, 3.91829361598142905208784365588, 3.93075915307069635047939476288, 4.25140047878725334557975372134, 4.35512192545127323722596941636, 5.01603688523649070095620499664, 5.24890504753722217841713238811, 5.41536720439755899801802541230, 5.56439446468880103450287802375, 5.99235413849677190310957446535, 6.27683672093954279489346229506, 6.36213954901504590093555987082, 6.52087869262768544828296409576, 6.55360044949997774916667933792
