| L(s) = 1 | − 4-s + 4·5-s + 16-s + 16·19-s − 4·20-s + 11·25-s + 16·29-s + 8·31-s − 24·41-s − 49-s + 16·59-s − 12·61-s − 64-s − 16·76-s − 16·79-s + 4·80-s − 8·89-s + 64·95-s − 11·100-s − 24·101-s − 4·109-s − 16·116-s − 22·121-s − 8·124-s + 24·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s + 1/4·16-s + 3.67·19-s − 0.894·20-s + 11/5·25-s + 2.97·29-s + 1.43·31-s − 3.74·41-s − 1/7·49-s + 2.08·59-s − 1.53·61-s − 1/8·64-s − 1.83·76-s − 1.80·79-s + 0.447·80-s − 0.847·89-s + 6.56·95-s − 1.09·100-s − 2.38·101-s − 0.383·109-s − 1.48·116-s − 2·121-s − 0.718·124-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.867934113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.867934113\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.31529745828327732583015865858, −10.28407581988392245336394221870, −9.941061038410829275019050461368, −9.748709981893755748827259663325, −9.036282714935392713844527018644, −8.871692668389388938023742599383, −8.092437036545450606450883741632, −8.029479390995284509235287394639, −6.97825568682782207222386409766, −6.85270210861731788899936460411, −6.37258544847786355920627883817, −5.67725156428613387770483250389, −5.22957916698797422259489754979, −5.09090846982921197241254871485, −4.52203207450589931304947774670, −3.52409716305951997408299658200, −2.89000593295966485594978736565, −2.72103708481317336112328359056, −1.35996701240880706455183564253, −1.19424583573088466237925119486,
1.19424583573088466237925119486, 1.35996701240880706455183564253, 2.72103708481317336112328359056, 2.89000593295966485594978736565, 3.52409716305951997408299658200, 4.52203207450589931304947774670, 5.09090846982921197241254871485, 5.22957916698797422259489754979, 5.67725156428613387770483250389, 6.37258544847786355920627883817, 6.85270210861731788899936460411, 6.97825568682782207222386409766, 8.029479390995284509235287394639, 8.092437036545450606450883741632, 8.871692668389388938023742599383, 9.036282714935392713844527018644, 9.748709981893755748827259663325, 9.941061038410829275019050461368, 10.28407581988392245336394221870, 10.31529745828327732583015865858
