| L(s) = 1 | − 1.93·2-s + 1.73·4-s + 1.12·7-s + 0.517·8-s − 1.59·11-s − 4.20·13-s − 2.17·14-s − 4.46·16-s + 1.41·17-s − 2·19-s + 3.07·22-s − 1.41·23-s + 8.11·26-s + 1.95·28-s + 1.00·29-s + 31-s + 7.58·32-s − 2.73·34-s + 5.02·37-s + 3.86·38-s + 5.94·41-s − 3.07·43-s − 2.75·44-s + 2.73·46-s + 8.10·47-s − 5.73·49-s − 7.27·52-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.866·4-s + 0.425·7-s + 0.183·8-s − 0.480·11-s − 1.16·13-s − 0.581·14-s − 1.11·16-s + 0.342·17-s − 0.458·19-s + 0.655·22-s − 0.294·23-s + 1.59·26-s + 0.368·28-s + 0.187·29-s + 0.179·31-s + 1.34·32-s − 0.468·34-s + 0.826·37-s + 0.626·38-s + 0.928·41-s − 0.469·43-s − 0.415·44-s + 0.402·46-s + 1.18·47-s − 0.818·49-s − 1.00·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 37 | \( 1 - 5.02T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 - 2.17T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 6.95T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
| 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
−7.78202425828023272431307569905, −7.23378628417000376197961009170, −6.47275770805022439527177873004, −5.50497682576887470012444909299, −4.76354748630309338410152183774, −4.06517650207814838609205077882, −2.72914796123447033383280434029, −2.11882465961510341252716643325, −1.06085314134552377309703716268, 0,
1.06085314134552377309703716268, 2.11882465961510341252716643325, 2.72914796123447033383280434029, 4.06517650207814838609205077882, 4.76354748630309338410152183774, 5.50497682576887470012444909299, 6.47275770805022439527177873004, 7.23378628417000376197961009170, 7.78202425828023272431307569905
