| L(s) = 1 | − 2-s + 4-s − 8-s − 4·9-s + 16-s + 4·18-s − 12·29-s − 32-s − 4·36-s − 12·37-s − 7·49-s − 12·53-s + 12·58-s + 64-s + 4·72-s + 12·74-s + 7·81-s + 7·98-s + 12·106-s + 4·109-s + 36·113-s − 12·116-s − 22·121-s + 127-s − 128-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 4/3·9-s + 1/4·16-s + 0.942·18-s − 2.22·29-s − 0.176·32-s − 2/3·36-s − 1.97·37-s − 49-s − 1.64·53-s + 1.57·58-s + 1/8·64-s + 0.471·72-s + 1.39·74-s + 7/9·81-s + 0.707·98-s + 1.16·106-s + 0.383·109-s + 3.38·113-s − 1.11·116-s − 2·121-s + 0.0887·127-s − 0.0883·128-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5466883037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5466883037\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.110658889510898732304582937668, −7.83907468238400266851940214611, −7.29352345837928406323819739763, −6.93521779634098499502560239431, −6.37212138742066738013944826790, −5.93784728662927089770286099022, −5.53261463807676054125487781511, −5.12538733530545440171177828433, −4.53924472595404667271186754504, −3.72028976828626557217306577048, −3.32856029125952418423355197897, −2.88347882202976614876074513114, −2.01306260891470067015653642288, −1.66378421862957417480655272149, −0.36742749720328815144260887896,
0.36742749720328815144260887896, 1.66378421862957417480655272149, 2.01306260891470067015653642288, 2.88347882202976614876074513114, 3.32856029125952418423355197897, 3.72028976828626557217306577048, 4.53924472595404667271186754504, 5.12538733530545440171177828433, 5.53261463807676054125487781511, 5.93784728662927089770286099022, 6.37212138742066738013944826790, 6.93521779634098499502560239431, 7.29352345837928406323819739763, 7.83907468238400266851940214611, 8.110658889510898732304582937668
