| L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·11-s − 2·13-s + 4·19-s + 2·21-s − 23-s − 5·25-s − 4·27-s + 6·29-s − 2·31-s − 8·33-s + 2·37-s − 4·39-s − 6·41-s − 4·43-s + 10·47-s + 49-s + 6·53-s + 8·57-s + 6·59-s − 4·61-s + 63-s − 4·67-s − 2·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.917·19-s + 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s + 1.11·29-s − 0.359·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 1.45·47-s + 1/7·49-s + 0.824·53-s + 1.05·57-s + 0.781·59-s − 0.512·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s − 0.949·71-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.86014537271703, −16.12721721776572, −15.63973260108531, −15.09334958712454, −14.63100236032693, −13.94011604540266, −13.54897787746538, −13.15086493320586, −12.12907220344678, −11.87536085298761, −10.98518015366856, −10.28496612782706, −9.825830988097532, −9.163512679778035, −8.461216886627752, −7.998123075616248, −7.511216491978507, −6.881784068896763, −5.723590035834086, −5.314412166325533, −4.432085603592244, −3.674197759076136, −2.787924486251786, −2.442679733706256, −1.432793377382518, 0,
1.432793377382518, 2.442679733706256, 2.787924486251786, 3.674197759076136, 4.432085603592244, 5.314412166325533, 5.723590035834086, 6.881784068896763, 7.511216491978507, 7.998123075616248, 8.461216886627752, 9.163512679778035, 9.825830988097532, 10.28496612782706, 10.98518015366856, 11.87536085298761, 12.12907220344678, 13.15086493320586, 13.54897787746538, 13.94011604540266, 14.63100236032693, 15.09334958712454, 15.63973260108531, 16.12721721776572, 16.86014537271703
