| L(s) = 1 | − 2.33·3-s − 5-s − 0.399·7-s + 2.43·9-s + 2.43·13-s + 2.33·15-s − 3.63·19-s + 0.932·21-s + 1.02·23-s + 25-s + 1.30·27-s + 4·29-s − 3.23·31-s + 0.399·35-s − 4.87·37-s − 5.68·39-s + 0.0922·41-s − 7.14·43-s − 2.43·45-s − 9.21·47-s − 6.84·49-s + 9.10·53-s + 8.48·57-s + 7.68·59-s + 6.57·61-s − 0.975·63-s − 2.43·65-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.447·5-s − 0.151·7-s + 0.813·9-s + 0.676·13-s + 0.602·15-s − 0.835·19-s + 0.203·21-s + 0.213·23-s + 0.200·25-s + 0.251·27-s + 0.742·29-s − 0.581·31-s + 0.0675·35-s − 0.802·37-s − 0.911·39-s + 0.0144·41-s − 1.08·43-s − 0.363·45-s − 1.34·47-s − 0.977·49-s + 1.25·53-s + 1.12·57-s + 1.00·59-s + 0.841·61-s − 0.122·63-s − 0.302·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 7 | \( 1 + 0.399T + 7T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 - 0.0922T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 9.10T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.28T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 1.67T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−6.90327798177255536451353566140, −6.76274931324364766431349381188, −5.98074434816839961478288564719, −5.29432827955211206983820061090, −4.73665502149639210799864936023, −3.91086508988740911223887757439, −3.19885794213389471721324091955, −2.00058167856979318063257307226, −0.935177326452036914089710265727, 0,
0.935177326452036914089710265727, 2.00058167856979318063257307226, 3.19885794213389471721324091955, 3.91086508988740911223887757439, 4.73665502149639210799864936023, 5.29432827955211206983820061090, 5.98074434816839961478288564719, 6.76274931324364766431349381188, 6.90327798177255536451353566140
