| L(s) = 1 | − 0.762·2-s − 1.41·4-s + 4.10·5-s − 2.37·7-s + 2.60·8-s − 3.12·10-s + 4.76·11-s + 5.19·13-s + 1.81·14-s + 0.848·16-s + 2.80·17-s − 0.644·19-s − 5.82·20-s − 3.63·22-s − 23-s + 11.8·25-s − 3.95·26-s + 3.36·28-s − 29-s + 0.195·31-s − 5.86·32-s − 2.14·34-s − 9.75·35-s + 4.46·37-s + 0.491·38-s + 10.6·40-s − 2.20·41-s + ⋯ |
| L(s) = 1 | − 0.539·2-s − 0.709·4-s + 1.83·5-s − 0.898·7-s + 0.921·8-s − 0.989·10-s + 1.43·11-s + 1.43·13-s + 0.484·14-s + 0.212·16-s + 0.681·17-s − 0.147·19-s − 1.30·20-s − 0.775·22-s − 0.208·23-s + 2.36·25-s − 0.776·26-s + 0.636·28-s − 0.185·29-s + 0.0351·31-s − 1.03·32-s − 0.367·34-s − 1.64·35-s + 0.734·37-s + 0.0797·38-s + 1.69·40-s − 0.343·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.174013421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174013421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.762T + 2T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 0.644T + 19T^{2} \) |
| 31 | \( 1 - 0.195T + 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 - 6.79T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 8.72T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| 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
−8.487929435468842245332724558295, −7.27302870526014106431533289743, −6.57149622266175429586111176312, −5.83069447502824361635168286869, −5.64084827273190011769408638033, −4.30972984123953668744033371632, −3.72213626239493189114413114556, −2.66377694224741364150162044108, −1.49258387590045177851589890559, −0.982871907352630648614859081544,
0.982871907352630648614859081544, 1.49258387590045177851589890559, 2.66377694224741364150162044108, 3.72213626239493189114413114556, 4.30972984123953668744033371632, 5.64084827273190011769408638033, 5.83069447502824361635168286869, 6.57149622266175429586111176312, 7.27302870526014106431533289743, 8.487929435468842245332724558295
