| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 2·13-s + 16-s − 2·17-s − 18-s − 8·19-s + 23-s − 24-s − 2·26-s + 27-s − 2·29-s − 8·31-s − 32-s + 2·34-s + 36-s − 2·37-s + 8·38-s + 2·39-s + 10·41-s − 8·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.208·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.328·37-s + 1.29·38-s + 0.320·39-s + 1.56·41-s − 1.21·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.209612442355213824636304538581, −7.77628585636150606263250003281, −6.71208026569885213236858813762, −6.33258448748158250980002056323, −5.19316738687988757960375439098, −4.17642925250835574770122239688, −3.39677865046872893785625566054, −2.31753412322329380076644813977, −1.57259474920656809409354647517, 0,
1.57259474920656809409354647517, 2.31753412322329380076644813977, 3.39677865046872893785625566054, 4.17642925250835574770122239688, 5.19316738687988757960375439098, 6.33258448748158250980002056323, 6.71208026569885213236858813762, 7.77628585636150606263250003281, 8.209612442355213824636304538581
