| L(s) = 1 | − 0.474·3-s + 1.63·5-s − 2.77·9-s − 0.241·11-s + 1.90·13-s − 0.775·15-s + 17-s − 5.11·19-s + 3.66·23-s − 2.32·25-s + 2.73·27-s + 2.50·29-s − 2.41·31-s + 0.114·33-s + 3.35·37-s − 0.902·39-s − 9.52·41-s + 9.54·43-s − 4.53·45-s − 8.63·47-s − 0.474·51-s − 11.6·53-s − 0.394·55-s + 2.42·57-s − 2.07·59-s + 10.6·61-s + 3.11·65-s + ⋯ |
| L(s) = 1 | − 0.273·3-s + 0.731·5-s − 0.924·9-s − 0.0726·11-s + 0.527·13-s − 0.200·15-s + 0.242·17-s − 1.17·19-s + 0.764·23-s − 0.465·25-s + 0.527·27-s + 0.464·29-s − 0.433·31-s + 0.0199·33-s + 0.551·37-s − 0.144·39-s − 1.48·41-s + 1.45·43-s − 0.676·45-s − 1.26·47-s − 0.0664·51-s − 1.59·53-s − 0.0531·55-s + 0.321·57-s − 0.269·59-s + 1.36·61-s + 0.386·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.474T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 11 | \( 1 + 0.241T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 9.52T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 + 8.63T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 0.576T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 8.46T + 79T^{2} \) |
| 83 | \( 1 - 1.02T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
| 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
−7.76974534124889555215287059289, −6.59528460324967074314081956632, −6.33016898390923616294319536510, −5.52565565512254815795327832198, −4.97432642672980333798658879065, −3.99376954166514568499387379033, −3.08695426353451776738378106320, −2.30075257737528629364590987850, −1.33336941965321098235892111349, 0,
1.33336941965321098235892111349, 2.30075257737528629364590987850, 3.08695426353451776738378106320, 3.99376954166514568499387379033, 4.97432642672980333798658879065, 5.52565565512254815795327832198, 6.33016898390923616294319536510, 6.59528460324967074314081956632, 7.76974534124889555215287059289
