| L(s) = 1 | − 3·2-s + 5·4-s − 7-s − 5·8-s − 9-s − 9·11-s + 3·14-s + 16-s + 3·18-s + 27·22-s − 7·23-s − 3·25-s − 5·28-s − 29-s + 7·32-s − 5·36-s + 14·37-s − 6·43-s − 45·44-s + 21·46-s − 6·49-s + 9·50-s − 53-s + 5·56-s + 3·58-s + 63-s − 15·64-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 5/2·4-s − 0.377·7-s − 1.76·8-s − 1/3·9-s − 2.71·11-s + 0.801·14-s + 1/4·16-s + 0.707·18-s + 5.75·22-s − 1.45·23-s − 3/5·25-s − 0.944·28-s − 0.185·29-s + 1.23·32-s − 5/6·36-s + 2.30·37-s − 0.914·43-s − 6.78·44-s + 3.09·46-s − 6/7·49-s + 1.27·50-s − 0.137·53-s + 0.668·56-s + 0.393·58-s + 0.125·63-s − 1.87·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 11 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
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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
−11.21366454432362178409800896144, −10.67031227449828284600588290559, −10.15719752606998884328845703933, −9.827750040154990690504559207296, −9.348106052753687446219355176798, −8.391620093074762865941497641656, −8.056074514521538854040464618150, −7.77684157811220129973739110796, −7.07843402085020079516919923744, −6.12517083143332380413032407232, −5.51689714943361794564222825282, −4.48039111841836744764506978027, −2.94840603147446336550026274361, −2.15224515309300368213938566990, 0,
2.15224515309300368213938566990, 2.94840603147446336550026274361, 4.48039111841836744764506978027, 5.51689714943361794564222825282, 6.12517083143332380413032407232, 7.07843402085020079516919923744, 7.77684157811220129973739110796, 8.056074514521538854040464618150, 8.391620093074762865941497641656, 9.348106052753687446219355176798, 9.827750040154990690504559207296, 10.15719752606998884328845703933, 10.67031227449828284600588290559, 11.21366454432362178409800896144
