| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 9-s − 2·10-s − 2·13-s + 16-s + 16·17-s + 18-s − 2·20-s + 3·25-s − 2·26-s − 8·29-s + 32-s + 16·34-s + 36-s − 4·37-s − 2·40-s − 4·41-s − 2·45-s − 10·49-s + 3·50-s − 2·52-s − 20·53-s − 8·58-s − 20·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.554·13-s + 1/4·16-s + 3.88·17-s + 0.235·18-s − 0.447·20-s + 3/5·25-s − 0.392·26-s − 1.48·29-s + 0.176·32-s + 2.74·34-s + 1/6·36-s − 0.657·37-s − 0.316·40-s − 0.624·41-s − 0.298·45-s − 1.42·49-s + 0.424·50-s − 0.277·52-s − 2.74·53-s − 1.05·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−7.76948487871725340041744632337, −7.38229162108116731403430084124, −7.09699921362048702400785248351, −6.46010665238686597416245482811, −5.88823502695794701483978541538, −5.36890034019942549244695495877, −5.35993903747378895770836082484, −4.46780194170664996451904659653, −4.28385424811650436847502480563, −3.39888206557913870378261957815, −3.22048752563716810372968934226, −2.96849062059581912835324119736, −1.52256173769279382883262599647, −1.48277331421431160180933642843, 0,
1.48277331421431160180933642843, 1.52256173769279382883262599647, 2.96849062059581912835324119736, 3.22048752563716810372968934226, 3.39888206557913870378261957815, 4.28385424811650436847502480563, 4.46780194170664996451904659653, 5.35993903747378895770836082484, 5.36890034019942549244695495877, 5.88823502695794701483978541538, 6.46010665238686597416245482811, 7.09699921362048702400785248351, 7.38229162108116731403430084124, 7.76948487871725340041744632337
