| L(s) = 1 | + 1.23·2-s − 0.470·4-s − 0.403·5-s − 0.303·7-s − 3.05·8-s − 0.498·10-s + 11-s − 3.53·13-s − 0.375·14-s − 2.83·16-s + 7.19·17-s + 3.67·19-s + 0.189·20-s + 1.23·22-s − 3.45·23-s − 4.83·25-s − 4.36·26-s + 0.142·28-s + 2.49·29-s + 2.86·31-s + 2.59·32-s + 8.90·34-s + 0.122·35-s + 3.00·37-s + 4.54·38-s + 1.23·40-s + 4.84·41-s + ⋯ |
| L(s) = 1 | + 0.874·2-s − 0.235·4-s − 0.180·5-s − 0.114·7-s − 1.08·8-s − 0.157·10-s + 0.301·11-s − 0.979·13-s − 0.100·14-s − 0.709·16-s + 1.74·17-s + 0.842·19-s + 0.0424·20-s + 0.263·22-s − 0.720·23-s − 0.967·25-s − 0.856·26-s + 0.0269·28-s + 0.462·29-s + 0.515·31-s + 0.459·32-s + 1.52·34-s + 0.0206·35-s + 0.494·37-s + 0.736·38-s + 0.194·40-s + 0.757·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 5 | \( 1 + 0.403T + 5T^{2} \) |
| 7 | \( 1 + 0.303T + 7T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + 8.13T + 47T^{2} \) |
| 53 | \( 1 - 1.91T + 53T^{2} \) |
| 59 | \( 1 - 1.68T + 59T^{2} \) |
| 67 | \( 1 + 0.0898T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 2.45T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
| 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
−7.78283042111944053164525506407, −6.92770835135494694179500239006, −6.03152176155575652301196996640, −5.52091257250708716547389247085, −4.80284963750819807465937909230, −4.09529876231688252190606785444, −3.32192364611258338151805679717, −2.71316256936608940922924987366, −1.34101307526898467233252153295, 0,
1.34101307526898467233252153295, 2.71316256936608940922924987366, 3.32192364611258338151805679717, 4.09529876231688252190606785444, 4.80284963750819807465937909230, 5.52091257250708716547389247085, 6.03152176155575652301196996640, 6.92770835135494694179500239006, 7.78283042111944053164525506407
