| L(s) = 1 | − 3-s + 3·5-s + 5·7-s + 3·9-s − 3·11-s − 8·13-s − 3·15-s + 7·19-s − 5·21-s − 6·23-s + 5·25-s − 8·27-s + 12·29-s + 10·31-s + 3·33-s + 15·35-s − 2·37-s + 8·39-s − 2·41-s − 8·43-s + 9·45-s − 12·47-s + 18·49-s + 6·53-s − 9·55-s − 7·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.88·7-s + 9-s − 0.904·11-s − 2.21·13-s − 0.774·15-s + 1.60·19-s − 1.09·21-s − 1.25·23-s + 25-s − 1.53·27-s + 2.22·29-s + 1.79·31-s + 0.522·33-s + 2.53·35-s − 0.328·37-s + 1.28·39-s − 0.312·41-s − 1.21·43-s + 1.34·45-s − 1.75·47-s + 18/7·49-s + 0.824·53-s − 1.21·55-s − 0.927·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.802088179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.802088179\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.19106632542635086340634989061, −9.745729731948265619225177278201, −9.480014696826783134729229895965, −8.698030150360916937617584052215, −8.270633291709258339055474741293, −7.87167810182071626439439879439, −7.63486063245304789731337065657, −7.06770172590634087964419016770, −6.78910354742494506476077676976, −6.07925906365315351098141303553, −5.73073252471695500013273618058, −5.00807182546260997785011896876, −5.00204516769489710156568694167, −4.76953954725935704611228466520, −4.16636081779854867182988028584, −3.10841682492313998810225076496, −2.59876736114822818325043109856, −1.98504048598254542901029208621, −1.65635820597520950498015914372, −0.77575176012664874582822881449,
0.77575176012664874582822881449, 1.65635820597520950498015914372, 1.98504048598254542901029208621, 2.59876736114822818325043109856, 3.10841682492313998810225076496, 4.16636081779854867182988028584, 4.76953954725935704611228466520, 5.00204516769489710156568694167, 5.00807182546260997785011896876, 5.73073252471695500013273618058, 6.07925906365315351098141303553, 6.78910354742494506476077676976, 7.06770172590634087964419016770, 7.63486063245304789731337065657, 7.87167810182071626439439879439, 8.270633291709258339055474741293, 8.698030150360916937617584052215, 9.480014696826783134729229895965, 9.745729731948265619225177278201, 10.19106632542635086340634989061
