| L(s) = 1 | − 0.285·2-s − 1.61·3-s − 1.91·4-s + 5-s + 0.460·6-s − 1.06·7-s + 1.11·8-s − 0.388·9-s − 0.285·10-s + 11-s + 3.10·12-s + 2.90·13-s + 0.304·14-s − 1.61·15-s + 3.51·16-s + 4.79·17-s + 0.110·18-s − 19-s − 1.91·20-s + 1.72·21-s − 0.285·22-s − 9.15·23-s − 1.80·24-s + 25-s − 0.828·26-s + 5.47·27-s + 2.04·28-s + ⋯ |
| L(s) = 1 | − 0.201·2-s − 0.932·3-s − 0.959·4-s + 0.447·5-s + 0.188·6-s − 0.403·7-s + 0.395·8-s − 0.129·9-s − 0.0901·10-s + 0.301·11-s + 0.894·12-s + 0.806·13-s + 0.0813·14-s − 0.417·15-s + 0.879·16-s + 1.16·17-s + 0.0261·18-s − 0.229·19-s − 0.429·20-s + 0.376·21-s − 0.0608·22-s − 1.90·23-s − 0.368·24-s + 0.200·25-s − 0.162·26-s + 1.05·27-s + 0.386·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.285T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 23 | \( 1 + 9.15T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 6.45T + 43T^{2} \) |
| 47 | \( 1 + 2.00T + 47T^{2} \) |
| 53 | \( 1 + 8.33T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 + 0.940T + 67T^{2} \) |
| 71 | \( 1 + 2.06T + 71T^{2} \) |
| 73 | \( 1 + 4.63T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
| 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
−9.742213639191594128086413840357, −8.662892527821334104932278447158, −8.079907824870499110788891543154, −6.77382545330735325767164140387, −5.86107108288607702216405824363, −5.43793085898047789368980171558, −4.26700322673397745072295463249, −3.30556255162046587327841822211, −1.44795892180908466660114950798, 0,
1.44795892180908466660114950798, 3.30556255162046587327841822211, 4.26700322673397745072295463249, 5.43793085898047789368980171558, 5.86107108288607702216405824363, 6.77382545330735325767164140387, 8.079907824870499110788891543154, 8.662892527821334104932278447158, 9.742213639191594128086413840357
