L(s) = 1 | + 2-s + 4-s + 1.20·5-s − 4.29·7-s + 8-s + 1.20·10-s − 1.67·11-s − 2.77·13-s − 4.29·14-s + 16-s + 5.75·17-s + 5.66·19-s + 1.20·20-s − 1.67·22-s + 3.89·23-s − 3.55·25-s − 2.77·26-s − 4.29·28-s + 3.66·29-s − 10.2·31-s + 32-s + 5.75·34-s − 5.15·35-s − 7.55·37-s + 5.66·38-s + 1.20·40-s + 1.38·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.537·5-s − 1.62·7-s + 0.353·8-s + 0.379·10-s − 0.504·11-s − 0.768·13-s − 1.14·14-s + 0.250·16-s + 1.39·17-s + 1.30·19-s + 0.268·20-s − 0.356·22-s + 0.812·23-s − 0.711·25-s − 0.543·26-s − 0.811·28-s + 0.680·29-s − 1.83·31-s + 0.176·32-s + 0.986·34-s − 0.871·35-s − 1.24·37-s + 0.919·38-s + 0.189·40-s + 0.216·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 - 5.66T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 3.66T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 - 1.38T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 + 5.85T + 47T^{2} \) |
| 53 | \( 1 - 0.209T + 53T^{2} \) |
| 59 | \( 1 + 1.94T + 59T^{2} \) |
| 61 | \( 1 - 1.99T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 9.26T + 71T^{2} \) |
| 73 | \( 1 - 7.84T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.69T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24071540781687571561326182876, −6.81802431967281945907808970908, −5.89846661206055972107092607321, −5.46394954787974102894810228778, −4.88434189252307790539489226073, −3.55098899063089415008813826122, −3.29165131870790889709504733157, −2.51893023383560171048448915351, −1.40513406971607747537234216280, 0,
1.40513406971607747537234216280, 2.51893023383560171048448915351, 3.29165131870790889709504733157, 3.55098899063089415008813826122, 4.88434189252307790539489226073, 5.46394954787974102894810228778, 5.89846661206055972107092607321, 6.81802431967281945907808970908, 7.24071540781687571561326182876