| L(s) = 1 | − 1.22·2-s − 2.28·3-s − 0.507·4-s + 2.79·6-s + 1.28·7-s + 3.06·8-s + 2.22·9-s + 0.285·11-s + 1.15·12-s + 5·13-s − 1.57·14-s − 2.72·16-s + 6.23·17-s − 2.71·18-s − 2.93·21-s − 0.348·22-s + 5.23·23-s − 7.00·24-s − 6.10·26-s + 1.77·27-s − 0.651·28-s + 1.28·29-s − 1.22·31-s − 2.79·32-s − 0.651·33-s − 7.61·34-s − 1.12·36-s + ⋯ |
| L(s) = 1 | − 0.863·2-s − 1.31·3-s − 0.253·4-s + 1.13·6-s + 0.485·7-s + 1.08·8-s + 0.740·9-s + 0.0859·11-s + 0.334·12-s + 1.38·13-s − 0.419·14-s − 0.682·16-s + 1.51·17-s − 0.639·18-s − 0.640·21-s − 0.0742·22-s + 1.09·23-s − 1.42·24-s − 1.19·26-s + 0.342·27-s − 0.123·28-s + 0.238·29-s − 0.219·31-s − 0.493·32-s − 0.113·33-s − 1.30·34-s − 0.187·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 3 | \( 1 + 2.28T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 - 0.285T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 0.841T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 5.72T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 - 0.985T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 0.764T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 1.66T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
| 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
−7.37935871280600880556675927505, −6.79157274104391720954758943437, −5.97004284054082084866309174504, −5.32456062439040688562266779546, −4.87354510395221636648817183371, −3.95761720760126928031861548143, −3.13323321096127924486087134418, −1.46012659344141960234756797300, −1.15567198284948407463738011938, 0,
1.15567198284948407463738011938, 1.46012659344141960234756797300, 3.13323321096127924486087134418, 3.95761720760126928031861548143, 4.87354510395221636648817183371, 5.32456062439040688562266779546, 5.97004284054082084866309174504, 6.79157274104391720954758943437, 7.37935871280600880556675927505
