| L(s) = 1 | + 32·2-s + 108·3-s + 640·4-s + 270·5-s + 3.45e3·6-s + 2.02e3·7-s + 1.02e4·8-s + 7.29e3·9-s + 8.64e3·10-s + 4.12e3·11-s + 6.91e4·12-s + 8.03e3·13-s + 6.47e4·14-s + 2.91e4·15-s + 1.43e5·16-s + 3.71e4·17-s + 2.33e5·18-s + 5.70e3·19-s + 1.72e5·20-s + 2.18e5·21-s + 1.31e5·22-s − 4.86e4·23-s + 1.10e6·24-s − 5.90e4·25-s + 2.57e5·26-s + 3.93e5·27-s + 1.29e6·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s + 0.965·5-s + 6.53·6-s + 2.22·7-s + 7.07·8-s + 10/3·9-s + 2.73·10-s + 0.933·11-s + 11.5·12-s + 1.01·13-s + 6.30·14-s + 2.23·15-s + 35/4·16-s + 1.83·17-s + 9.42·18-s + 0.190·19-s + 4.82·20-s + 5.14·21-s + 2.63·22-s − 0.834·23-s + 16.3·24-s − 0.755·25-s + 2.86·26-s + 3.84·27-s + 11.1·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(501.8136018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(501.8136018\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 23 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 54 p T + 26392 p T^{2} - 92426 p^{3} T^{3} + 71603766 p^{3} T^{4} - 92426 p^{10} T^{5} + 26392 p^{15} T^{6} - 54 p^{22} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 2022 T + 3846604 T^{2} - 4295138534 T^{3} + 677209178010 p T^{4} - 4295138534 p^{7} T^{5} + 3846604 p^{14} T^{6} - 2022 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 4120 T + 68308716 T^{2} - 222162400920 T^{3} + 175272570086690 p T^{4} - 222162400920 p^{7} T^{5} + 68308716 p^{14} T^{6} - 4120 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 8036 T + 14886964 p T^{2} - 92788577212 p T^{3} + 17415004623051494 T^{4} - 92788577212 p^{8} T^{5} + 14886964 p^{15} T^{6} - 8036 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 37182 T + 1280173232 T^{2} - 31379975039338 T^{3} + 644580631532453406 T^{4} - 31379975039338 p^{7} T^{5} + 1280173232 p^{14} T^{6} - 37182 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 5702 T + 1220992612 T^{2} - 8279723258974 T^{3} + 95675377655176082 p T^{4} - 8279723258974 p^{7} T^{5} + 1220992612 p^{14} T^{6} - 5702 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 217716 T + 27997902692 T^{2} - 2295515593799420 T^{3} + \)\(17\!\cdots\!26\)\( T^{4} - 2295515593799420 p^{7} T^{5} + 27997902692 p^{14} T^{6} - 217716 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 222852 T + 67862099100 T^{2} - 9249924944188596 T^{3} + \)\(21\!\cdots\!58\)\( T^{4} - 9249924944188596 p^{7} T^{5} + 67862099100 p^{14} T^{6} - 222852 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 486428 T + 146937566836 T^{2} - 21472423287173588 T^{3} + \)\(72\!\cdots\!70\)\( T^{4} - 21472423287173588 p^{7} T^{5} + 146937566836 p^{14} T^{6} - 486428 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 338336 T + 17605077484 p T^{2} - 187441215406521888 T^{3} + \)\(20\!\cdots\!58\)\( T^{4} - 187441215406521888 p^{7} T^{5} + 17605077484 p^{15} T^{6} - 338336 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 730974 T + 811762626020 T^{2} - 441053942925665094 T^{3} + \)\(29\!\cdots\!82\)\( T^{4} - 441053942925665094 p^{7} T^{5} + 811762626020 p^{14} T^{6} - 730974 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 338248 T + 187675926092 T^{2} + 410886908939936856 T^{3} - \)\(68\!\cdots\!70\)\( T^{4} + 410886908939936856 p^{7} T^{5} + 187675926092 p^{14} T^{6} - 338248 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 375502 T + 2122636650016 T^{2} + 1230135410249443682 T^{3} + \)\(35\!\cdots\!66\)\( T^{4} + 1230135410249443682 p^{7} T^{5} + 2122636650016 p^{14} T^{6} + 375502 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 71392 T - 662841515140 T^{2} - 2118841469757327840 T^{3} + \)\(80\!\cdots\!86\)\( T^{4} - 2118841469757327840 p^{7} T^{5} - 662841515140 p^{14} T^{6} - 71392 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 2101164 T + 3834363672980 T^{2} - 7716262121001291684 T^{3} + \)\(22\!\cdots\!50\)\( T^{4} - 7716262121001291684 p^{7} T^{5} + 3834363672980 p^{14} T^{6} - 2101164 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 4337162 T + 14477665501876 T^{2} - 50034399154209618482 T^{3} + \)\(14\!\cdots\!30\)\( T^{4} - 50034399154209618482 p^{7} T^{5} + 14477665501876 p^{14} T^{6} - 4337162 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 2288016 T + 16671405535196 T^{2} - 34178229277045146576 T^{3} + \)\(23\!\cdots\!02\)\( T^{4} - 34178229277045146576 p^{7} T^{5} + 16671405535196 p^{14} T^{6} - 2288016 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1107328 T + 37898756072332 T^{2} + 30175989822481312704 T^{3} + \)\(59\!\cdots\!02\)\( T^{4} + 30175989822481312704 p^{7} T^{5} + 37898756072332 p^{14} T^{6} + 1107328 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 60610 T + 43736287090316 T^{2} - 32940296215812940242 T^{3} + \)\(10\!\cdots\!74\)\( T^{4} - 32940296215812940242 p^{7} T^{5} + 43736287090316 p^{14} T^{6} - 60610 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1485464 T + 61665606940204 T^{2} - \)\(15\!\cdots\!52\)\( T^{3} + \)\(19\!\cdots\!66\)\( T^{4} - \)\(15\!\cdots\!52\)\( p^{7} T^{5} + 61665606940204 p^{14} T^{6} - 1485464 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 1485090 T + 161051275514840 T^{2} - \)\(16\!\cdots\!38\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(16\!\cdots\!38\)\( p^{7} T^{5} + 161051275514840 p^{14} T^{6} - 1485090 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1935444 T + 123586624521636 T^{2} - \)\(29\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!58\)\( T^{4} - \)\(29\!\cdots\!96\)\( p^{7} T^{5} + 123586624521636 p^{14} T^{6} - 1935444 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−8.236651857478012138427357240126, −7.78061082974920519942276205052, −7.70633699156495431572487492622, −7.49614518543419847573736020138, −7.38246247312100045202263473206, −6.48664664178242141265545405314, −6.29134115546031098007449671901, −6.23944945264349408458013697014, −6.17555348447449859871475694932, −5.34355011466900104827637636938, −5.10143329624921625261834535948, −4.99547553315877963488143416291, −4.77283766144614629739974144027, −3.98357776240550575233263389675, −3.92244545264270325534484283473, −3.89757432786131582512881333811, −3.67277670272364354873462432855, −2.82960309484313075714797644820, −2.59994987819479963386236607898, −2.55535905929611009953205279110, −2.23926123181204487153841145198, −1.56059863423290988715566080699, −1.31951532557033063465385384888, −1.25749473667986984974774172860, −0.989916242951005570537731318052,
0.989916242951005570537731318052, 1.25749473667986984974774172860, 1.31951532557033063465385384888, 1.56059863423290988715566080699, 2.23926123181204487153841145198, 2.55535905929611009953205279110, 2.59994987819479963386236607898, 2.82960309484313075714797644820, 3.67277670272364354873462432855, 3.89757432786131582512881333811, 3.92244545264270325534484283473, 3.98357776240550575233263389675, 4.77283766144614629739974144027, 4.99547553315877963488143416291, 5.10143329624921625261834535948, 5.34355011466900104827637636938, 6.17555348447449859871475694932, 6.23944945264349408458013697014, 6.29134115546031098007449671901, 6.48664664178242141265545405314, 7.38246247312100045202263473206, 7.49614518543419847573736020138, 7.70633699156495431572487492622, 7.78061082974920519942276205052, 8.236651857478012138427357240126
