| L(s) = 1 | + 4-s − 4·5-s − 5·9-s + 4·11-s + 16-s − 2·19-s − 4·20-s + 11·25-s − 10·29-s − 16·31-s − 5·36-s − 16·41-s + 4·44-s + 20·45-s − 5·49-s − 16·55-s + 30·59-s + 4·61-s + 64-s + 4·71-s − 2·76-s − 20·79-s − 4·80-s + 16·81-s + 8·95-s − 20·99-s + 11·100-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s − 5/3·9-s + 1.20·11-s + 1/4·16-s − 0.458·19-s − 0.894·20-s + 11/5·25-s − 1.85·29-s − 2.87·31-s − 5/6·36-s − 2.49·41-s + 0.603·44-s + 2.98·45-s − 5/7·49-s − 2.15·55-s + 3.90·59-s + 0.512·61-s + 1/8·64-s + 0.474·71-s − 0.229·76-s − 2.25·79-s − 0.447·80-s + 16/9·81-s + 0.820·95-s − 2.01·99-s + 1.09·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.23294535946651466706548802912, −9.400115216603937442308106741391, −8.925019181414506634794762446017, −8.408486909011940238981299566497, −8.187366436155753750941904167188, −7.25038064772684706053554512263, −7.08084872197682116999638941334, −6.43803628048625394061789390103, −5.47809446086921756820160858625, −5.28069074518443602737935578596, −3.86057015395193111701613139713, −3.84969924867073870114275032751, −3.08844310830142045553246780849, −1.90909315273381969795467705091, 0,
1.90909315273381969795467705091, 3.08844310830142045553246780849, 3.84969924867073870114275032751, 3.86057015395193111701613139713, 5.28069074518443602737935578596, 5.47809446086921756820160858625, 6.43803628048625394061789390103, 7.08084872197682116999638941334, 7.25038064772684706053554512263, 8.187366436155753750941904167188, 8.408486909011940238981299566497, 8.925019181414506634794762446017, 9.400115216603937442308106741391, 10.23294535946651466706548802912
