| L(s) = 1 | + 21·5-s − 15·7-s − 39·11-s − 43·13-s + 27·23-s + 169·25-s + 165·29-s − 405·31-s − 315·35-s − 860·37-s + 369·41-s + 333·43-s + 33·47-s − 193·49-s − 819·55-s − 825·59-s + 745·61-s − 903·65-s − 789·67-s − 408·71-s + 428·73-s + 585·77-s − 1.01e3·79-s − 843·83-s + 645·91-s − 883·97-s + 1.88e3·101-s + ⋯ |
| L(s) = 1 | + 1.87·5-s − 0.809·7-s − 1.06·11-s − 0.917·13-s + 0.244·23-s + 1.35·25-s + 1.05·29-s − 2.34·31-s − 1.52·35-s − 3.82·37-s + 1.40·41-s + 1.18·43-s + 0.102·47-s − 0.562·49-s − 2.00·55-s − 1.82·59-s + 1.56·61-s − 1.72·65-s − 1.43·67-s − 0.681·71-s + 0.686·73-s + 0.865·77-s − 1.43·79-s − 1.11·83-s + 0.743·91-s − 0.924·97-s + 1.85·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.500880729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500880729\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 21 T + 272 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 15 T + 418 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 39 T + 190 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 43 T - 348 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1714 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2186 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 27 T - 11438 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 165 T + 33464 T^{2} - 165 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 405 T + 84466 T^{2} + 405 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 430 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 p T + 68 p^{2} T^{2} - 9 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 333 T + 116470 T^{2} - 333 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 33 T - 102734 T^{2} - 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 48922 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 825 T + 475246 T^{2} + 825 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 745 T + 328044 T^{2} - 745 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 789 T + 508270 T^{2} + 789 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1011 T + 833746 T^{2} + 1011 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 843 T + 138862 T^{2} + 843 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 568094 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 883 T - 132984 T^{2} + 883 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.75672778717380705911082905448, −10.29841277092589386784579996080, −10.20129918703400602383290719674, −9.573434921365480682684806936668, −9.320497229136884257044801076199, −8.823746948060577728565859271468, −8.427798960709711476689930259516, −7.45941172874442191647446925903, −7.29129861927152933951319916217, −6.78747714875566957081943646450, −6.07506697032771257749161361318, −5.76181472847648650100506230008, −5.30469577410016459589255774716, −4.93159795926216066772256720172, −4.10123930463468362420120576294, −3.16118315367952411609231980048, −2.85618842990149769263534382828, −1.98636126199103813770370413284, −1.72331705155652266862684240024, −0.36990702462857795850342243680,
0.36990702462857795850342243680, 1.72331705155652266862684240024, 1.98636126199103813770370413284, 2.85618842990149769263534382828, 3.16118315367952411609231980048, 4.10123930463468362420120576294, 4.93159795926216066772256720172, 5.30469577410016459589255774716, 5.76181472847648650100506230008, 6.07506697032771257749161361318, 6.78747714875566957081943646450, 7.29129861927152933951319916217, 7.45941172874442191647446925903, 8.427798960709711476689930259516, 8.823746948060577728565859271468, 9.320497229136884257044801076199, 9.573434921365480682684806936668, 10.20129918703400602383290719674, 10.29841277092589386784579996080, 10.75672778717380705911082905448
