L(s) = 1 | − 2-s − 0.936·3-s + 4-s + 1.38·5-s + 0.936·6-s + 2.49·7-s − 8-s − 2.12·9-s − 1.38·10-s − 1.29·11-s − 0.936·12-s − 0.532·13-s − 2.49·14-s − 1.29·15-s + 16-s − 4.49·17-s + 2.12·18-s − 6.56·19-s + 1.38·20-s − 2.33·21-s + 1.29·22-s + 23-s + 0.936·24-s − 3.09·25-s + 0.532·26-s + 4.79·27-s + 2.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.540·3-s + 0.5·4-s + 0.617·5-s + 0.382·6-s + 0.942·7-s − 0.353·8-s − 0.707·9-s − 0.436·10-s − 0.390·11-s − 0.270·12-s − 0.147·13-s − 0.666·14-s − 0.334·15-s + 0.250·16-s − 1.08·17-s + 0.500·18-s − 1.50·19-s + 0.308·20-s − 0.509·21-s + 0.276·22-s + 0.208·23-s + 0.191·24-s − 0.618·25-s + 0.104·26-s + 0.923·27-s + 0.471·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 0.936T + 3T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 + 0.532T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 - 5.61T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 - 0.607T + 53T^{2} \) |
| 59 | \( 1 + 0.436T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 8.74T + 67T^{2} \) |
| 71 | \( 1 - 3.67T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 9.05T + 79T^{2} \) |
| 83 | \( 1 + 7.36T + 83T^{2} \) |
| 89 | \( 1 + 7.67T + 89T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.923857932836746890349501553683, −6.96496992978136593881048500021, −6.22726704494184958717026206847, −5.85431334459629639685042662028, −4.78850379481669244458073408103, −4.37602129273523110006669330162, −2.74632354877596414511034921804, −2.30437524516728822387517817983, −1.20674295107536777458686810796, 0,
1.20674295107536777458686810796, 2.30437524516728822387517817983, 2.74632354877596414511034921804, 4.37602129273523110006669330162, 4.78850379481669244458073408103, 5.85431334459629639685042662028, 6.22726704494184958717026206847, 6.96496992978136593881048500021, 7.923857932836746890349501553683