| L(s) = 1 | − 0.947·3-s − 3.46·5-s − 2.10·9-s + 4.20·11-s − 1.85·13-s + 3.27·15-s + 0.291·17-s − 4.29·19-s + 23-s + 6.97·25-s + 4.83·27-s + 5.69·29-s − 2.11·31-s − 3.98·33-s − 10.2·37-s + 1.76·39-s − 9.06·41-s + 3.38·43-s + 7.27·45-s + 10.1·47-s − 0.276·51-s + 1.74·53-s − 14.5·55-s + 4.07·57-s + 2.45·59-s − 1.74·61-s + 6.43·65-s + ⋯ |
| L(s) = 1 | − 0.546·3-s − 1.54·5-s − 0.701·9-s + 1.26·11-s − 0.515·13-s + 0.846·15-s + 0.0707·17-s − 0.986·19-s + 0.208·23-s + 1.39·25-s + 0.930·27-s + 1.05·29-s − 0.379·31-s − 0.693·33-s − 1.68·37-s + 0.282·39-s − 1.41·41-s + 0.516·43-s + 1.08·45-s + 1.47·47-s − 0.0386·51-s + 0.239·53-s − 1.96·55-s + 0.539·57-s + 0.319·59-s − 0.224·61-s + 0.798·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.947T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 - 0.291T + 17T^{2} \) |
| 19 | \( 1 + 4.29T + 19T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.06T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.74T + 53T^{2} \) |
| 59 | \( 1 - 2.45T + 59T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 - 7.42T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 8.50T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 2.10T + 89T^{2} \) |
| 97 | \( 1 - 2.71T + 97T^{2} \) |
| 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
−7.26589287017088818904061731231, −6.78069455720711235201550587186, −6.15599744306564171626411073058, −5.19257725164604710188257565820, −4.59398660153273224801191630531, −3.82858370608814498856736295812, −3.31553041036893682730813401390, −2.21691288525934174643064421702, −0.905840250782146102667640843914, 0,
0.905840250782146102667640843914, 2.21691288525934174643064421702, 3.31553041036893682730813401390, 3.82858370608814498856736295812, 4.59398660153273224801191630531, 5.19257725164604710188257565820, 6.15599744306564171626411073058, 6.78069455720711235201550587186, 7.26589287017088818904061731231
