| L(s) = 1 | − 1.88·2-s + 1.53·4-s − 1.27·5-s + 0.0298·7-s + 0.873·8-s + 2.40·10-s + 11-s + 5.58·13-s − 0.0560·14-s − 4.71·16-s − 1.33·17-s + 6.70·19-s − 1.96·20-s − 1.88·22-s − 3.41·23-s − 3.36·25-s − 10.5·26-s + 0.0457·28-s − 1.60·29-s − 6.25·31-s + 7.11·32-s + 2.51·34-s − 0.0381·35-s + 1.71·37-s − 12.6·38-s − 1.11·40-s − 0.668·41-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.767·4-s − 0.571·5-s + 0.0112·7-s + 0.308·8-s + 0.760·10-s + 0.301·11-s + 1.54·13-s − 0.0149·14-s − 1.17·16-s − 0.324·17-s + 1.53·19-s − 0.438·20-s − 0.400·22-s − 0.711·23-s − 0.673·25-s − 2.06·26-s + 0.00865·28-s − 0.298·29-s − 1.12·31-s + 1.25·32-s + 0.431·34-s − 0.00644·35-s + 0.282·37-s − 2.04·38-s − 0.176·40-s − 0.104·41-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 | \( 1 + 1.88T + 2T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 7 | \( 1 - 0.0298T + 7T^{2} \) |
| 13 | \( 1 - 5.58T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 - 1.71T + 37T^{2} \) |
| 41 | \( 1 + 0.668T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 - 0.587T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 2.03T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 - 0.742T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 1.56T + 97T^{2} \) |
| 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
−7.77003711294318593425391421287, −7.44294591967731931891805873024, −6.45686354683732454529338704638, −5.82345000997886580111097604551, −4.76078422282104934953257398082, −3.89700296186266700595996953066, −3.25383491959974733879555945498, −1.86855638297612403874351645967, −1.15739366113225922690346739511, 0,
1.15739366113225922690346739511, 1.86855638297612403874351645967, 3.25383491959974733879555945498, 3.89700296186266700595996953066, 4.76078422282104934953257398082, 5.82345000997886580111097604551, 6.45686354683732454529338704638, 7.44294591967731931891805873024, 7.77003711294318593425391421287
