| L(s) = 1 | + 2·3-s − 4·5-s − 7-s + 9-s + 4·11-s + 2·13-s − 8·15-s − 4·17-s − 2·21-s + 23-s + 11·25-s − 4·27-s − 2·29-s + 6·31-s + 8·33-s + 4·35-s − 2·37-s + 4·39-s − 6·41-s − 4·43-s − 4·45-s + 10·47-s + 49-s − 8·51-s − 6·53-s − 16·55-s − 2·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 2.06·15-s − 0.970·17-s − 0.436·21-s + 0.208·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s + 1.07·31-s + 1.39·33-s + 0.676·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.596·45-s + 1.45·47-s + 1/7·49-s − 1.12·51-s − 0.824·53-s − 2.15·55-s − 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−16.94676360329807, −16.03269902423553, −15.61845731427810, −15.13780755123296, −14.82675174382004, −13.90310929179573, −13.70566227573478, −12.83960485409212, −12.18080688981024, −11.72966760645479, −11.14732092847532, −10.59356843280169, −9.537107253538659, −9.043717908515607, −8.532392917024111, −8.095828794558338, −7.418788672335999, −6.766472042880628, −6.236409020911850, −4.949794556786621, −4.198115168219906, −3.683112753245586, −3.263044139768560, −2.359317173714747, −1.192237831358074, 0,
1.192237831358074, 2.359317173714747, 3.263044139768560, 3.683112753245586, 4.198115168219906, 4.949794556786621, 6.236409020911850, 6.766472042880628, 7.418788672335999, 8.095828794558338, 8.532392917024111, 9.043717908515607, 9.537107253538659, 10.59356843280169, 11.14732092847532, 11.72966760645479, 12.18080688981024, 12.83960485409212, 13.70566227573478, 13.90310929179573, 14.82675174382004, 15.13780755123296, 15.61845731427810, 16.03269902423553, 16.94676360329807
