| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 4·7-s − 8·8-s − 10·10-s − 42·11-s + 20·13-s + 8·14-s + 16·16-s − 93·17-s + 59·19-s + 20·20-s + 84·22-s − 9·23-s + 25·25-s − 40·26-s − 16·28-s − 120·29-s + 47·31-s − 32·32-s + 186·34-s − 20·35-s − 262·37-s − 118·38-s − 40·40-s − 126·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.215·7-s − 0.353·8-s − 0.316·10-s − 1.15·11-s + 0.426·13-s + 0.152·14-s + 1/4·16-s − 1.32·17-s + 0.712·19-s + 0.223·20-s + 0.814·22-s − 0.0815·23-s + 1/5·25-s − 0.301·26-s − 0.107·28-s − 0.768·29-s + 0.272·31-s − 0.176·32-s + 0.938·34-s − 0.0965·35-s − 1.16·37-s − 0.503·38-s − 0.158·40-s − 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 93 T + p^{3} T^{2} \) |
| 19 | \( 1 - 59 T + p^{3} T^{2} \) |
| 23 | \( 1 + 9 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 47 T + p^{3} T^{2} \) |
| 37 | \( 1 + 262 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 + 741 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 221 T + p^{3} T^{2} \) |
| 67 | \( 1 + 538 T + p^{3} T^{2} \) |
| 71 | \( 1 + 690 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1126 T + p^{3} T^{2} \) |
| 79 | \( 1 - 665 T + p^{3} T^{2} \) |
| 83 | \( 1 + 75 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1086 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1544 T + p^{3} T^{2} \) |
| 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
−10.82387933032610466159230458133, −10.05188959390701941730898889448, −9.110335073730831573137237992067, −8.209429919550453124633346337166, −7.13752066718307239842628501035, −6.10908552591613562674787956181, −4.94045888314730274073260402257, −3.16781717532362284854674510984, −1.84338974082293675312061668138, 0,
1.84338974082293675312061668138, 3.16781717532362284854674510984, 4.94045888314730274073260402257, 6.10908552591613562674787956181, 7.13752066718307239842628501035, 8.209429919550453124633346337166, 9.110335073730831573137237992067, 10.05188959390701941730898889448, 10.82387933032610466159230458133
