| L(s) = 1 | − 2.73·3-s + 2.27·5-s − 0.610·7-s + 4.46·9-s + 0.399·11-s − 4.06·13-s − 6.22·15-s + 4.88·17-s − 5.94·19-s + 1.66·21-s + 1.42·23-s + 0.186·25-s − 3.99·27-s − 3.82·29-s + 31-s − 1.09·33-s − 1.38·35-s + 3.30·37-s + 11.1·39-s + 0.943·41-s − 9.82·43-s + 10.1·45-s + 3.70·47-s − 6.62·49-s − 13.3·51-s + 12.6·53-s + 0.909·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 1.01·5-s − 0.230·7-s + 1.48·9-s + 0.120·11-s − 1.12·13-s − 1.60·15-s + 1.18·17-s − 1.36·19-s + 0.363·21-s + 0.296·23-s + 0.0373·25-s − 0.769·27-s − 0.709·29-s + 0.179·31-s − 0.189·33-s − 0.234·35-s + 0.543·37-s + 1.77·39-s + 0.147·41-s − 1.49·43-s + 1.51·45-s + 0.540·47-s − 0.946·49-s − 1.86·51-s + 1.73·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 + 0.610T + 7T^{2} \) |
| 11 | \( 1 - 0.399T + 11T^{2} \) |
| 13 | \( 1 + 4.06T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 - 0.943T + 41T^{2} \) |
| 43 | \( 1 + 9.82T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 7.96T + 59T^{2} \) |
| 61 | \( 1 + 4.06T + 61T^{2} \) |
| 67 | \( 1 + 9.22T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 8.64T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
| 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
−9.001708448907781883429513975254, −7.80984055368228142406701929121, −6.89580969677615240479230846210, −6.28072513747326320377107641135, −5.56993289510701551414059799366, −5.05750384479045826220815045153, −4.04965481689797738659761446732, −2.58100512425161424347475420584, −1.42762259418391195036813435080, 0,
1.42762259418391195036813435080, 2.58100512425161424347475420584, 4.04965481689797738659761446732, 5.05750384479045826220815045153, 5.56993289510701551414059799366, 6.28072513747326320377107641135, 6.89580969677615240479230846210, 7.80984055368228142406701929121, 9.001708448907781883429513975254
