L(s) = 1 | − 1.19·2-s + 1.69·3-s − 0.564·4-s + 5-s − 2.03·6-s − 2.11·7-s + 3.07·8-s − 0.122·9-s − 1.19·10-s + 11-s − 0.958·12-s + 2.55·13-s + 2.53·14-s + 1.69·15-s − 2.55·16-s − 0.152·17-s + 0.146·18-s − 3.06·19-s − 0.564·20-s − 3.58·21-s − 1.19·22-s − 0.305·23-s + 5.21·24-s + 25-s − 3.05·26-s − 5.29·27-s + 1.19·28-s + ⋯ |
L(s) = 1 | − 0.847·2-s + 0.979·3-s − 0.282·4-s + 0.447·5-s − 0.829·6-s − 0.799·7-s + 1.08·8-s − 0.0407·9-s − 0.378·10-s + 0.301·11-s − 0.276·12-s + 0.707·13-s + 0.677·14-s + 0.438·15-s − 0.637·16-s − 0.0369·17-s + 0.0345·18-s − 0.702·19-s − 0.126·20-s − 0.783·21-s − 0.255·22-s − 0.0636·23-s + 1.06·24-s + 0.200·25-s − 0.599·26-s − 1.01·27-s + 0.225·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 0.152T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 0.305T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 + 7.99T + 43T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 0.0404T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 79 | \( 1 - 5.07T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.222376233834642371559367197034, −7.72125159761717788110750287594, −6.69738640556912010110513191139, −6.07978913197615674836279736038, −5.05546666734349535072144526314, −3.99900471421972625119229918096, −3.37544258135345362048432167014, −2.34354009251685186504583937189, −1.41872449098681953959420128456, 0,
1.41872449098681953959420128456, 2.34354009251685186504583937189, 3.37544258135345362048432167014, 3.99900471421972625119229918096, 5.05546666734349535072144526314, 6.07978913197615674836279736038, 6.69738640556912010110513191139, 7.72125159761717788110750287594, 8.222376233834642371559367197034