| L(s) = 1 | − 4·5-s − 6·11-s − 2·13-s + 2·17-s + 19-s + 6·23-s + 5·25-s − 4·29-s − 20·31-s + 4·37-s + 9·41-s + 4·43-s − 12·47-s − 14·49-s − 2·53-s + 24·55-s − 59-s + 8·61-s + 8·65-s − 9·67-s − 6·71-s + 9·73-s + 4·79-s + 10·83-s − 8·85-s − 18·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.80·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 25-s − 0.742·29-s − 3.59·31-s + 0.657·37-s + 1.40·41-s + 0.609·43-s − 1.75·47-s − 2·49-s − 0.274·53-s + 3.23·55-s − 0.130·59-s + 1.02·61-s + 0.992·65-s − 1.09·67-s − 0.712·71-s + 1.05·73-s + 0.450·79-s + 1.09·83-s − 0.867·85-s − 1.90·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.457994125107719805097877520386, −9.107795273162901345864065081160, −8.216124615608848005661779157040, −8.200689517438082779594401408425, −7.72290427755242095321194105477, −7.45095371135005017210275298730, −7.22401073194221916980695950331, −6.71913611105134990652463654128, −5.98152341278566681831237526760, −5.43214842127881589586722106763, −5.20209298951459580357708473155, −4.81663906891376778980749438302, −4.11240635181886462407126773576, −3.78285456924514605925214456129, −3.24943920395507697830303122771, −2.85589525382821161172138339512, −2.19113075987656725040316578571, −1.34720092598821044414781378375, 0, 0,
1.34720092598821044414781378375, 2.19113075987656725040316578571, 2.85589525382821161172138339512, 3.24943920395507697830303122771, 3.78285456924514605925214456129, 4.11240635181886462407126773576, 4.81663906891376778980749438302, 5.20209298951459580357708473155, 5.43214842127881589586722106763, 5.98152341278566681831237526760, 6.71913611105134990652463654128, 7.22401073194221916980695950331, 7.45095371135005017210275298730, 7.72290427755242095321194105477, 8.200689517438082779594401408425, 8.216124615608848005661779157040, 9.107795273162901345864065081160, 9.457994125107719805097877520386
