| L(s) = 1 | − 0.936·3-s − 0.936·7-s − 2.12·9-s − 5.73·11-s + 1.12·13-s + 0.876·21-s + 9.47·23-s − 5·25-s + 4.79·27-s + 7.60·31-s + 5.36·33-s − 1.05·39-s − 6.12·49-s + 8.24·53-s + 1.98·63-s − 8.87·69-s + 14.2·71-s + 4.68·75-s + 5.36·77-s + 13.2·79-s + 1.87·81-s − 7.36·89-s − 1.05·91-s − 7.12·93-s + 12.1·99-s − 19.3·101-s + 16.1·107-s + ⋯ |
| L(s) = 1 | − 0.540·3-s − 0.353·7-s − 0.707·9-s − 1.72·11-s + 0.311·13-s + 0.191·21-s + 1.97·23-s − 25-s + 0.923·27-s + 1.36·31-s + 0.934·33-s − 0.168·39-s − 0.874·49-s + 1.13·53-s + 0.250·63-s − 1.06·69-s + 1.69·71-s + 0.540·75-s + 0.611·77-s + 1.48·79-s + 0.208·81-s − 0.781·89-s − 0.110·91-s − 0.738·93-s + 1.22·99-s − 1.92·101-s + 1.56·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 0.936T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 0.936T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7.60T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
| 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
−7.37826069202933612184872296737, −6.57166384321075983517455869964, −6.01225759760212278224065525675, −5.12424917598310310417686738549, −5.01648970852463866293968542496, −3.77097142506627524507116589954, −2.89997270936862979540674669358, −2.42909922368237619991298575715, −0.980297091292653286074102024689, 0,
0.980297091292653286074102024689, 2.42909922368237619991298575715, 2.89997270936862979540674669358, 3.77097142506627524507116589954, 5.01648970852463866293968542496, 5.12424917598310310417686738549, 6.01225759760212278224065525675, 6.57166384321075983517455869964, 7.37826069202933612184872296737
