| L(s) = 1 | + 3-s − 2·4-s + 3·5-s + 7-s − 2·9-s − 3·11-s − 2·12-s + 13-s + 3·15-s + 4·16-s − 6·17-s − 7·19-s − 6·20-s + 21-s − 23-s + 4·25-s − 5·27-s − 2·28-s − 6·29-s − 4·31-s − 3·33-s + 3·35-s + 4·36-s + 2·37-s + 39-s + 6·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 0.277·13-s + 0.774·15-s + 16-s − 1.45·17-s − 1.60·19-s − 1.34·20-s + 0.218·21-s − 0.208·23-s + 4/5·25-s − 0.962·27-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.522·33-s + 0.507·35-s + 2/3·36-s + 0.328·37-s + 0.160·39-s + 0.937·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2093 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2093 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−8.652084536908547434382936656098, −8.399892326776514295186608795368, −7.26044041738974169680212516423, −6.06653547310959621978636638368, −5.59314981161876502315638963320, −4.69605768016550390702718627944, −3.81090451199532401548258489148, −2.52461807731621320682392403115, −1.90178872858867228525064546197, 0,
1.90178872858867228525064546197, 2.52461807731621320682392403115, 3.81090451199532401548258489148, 4.69605768016550390702718627944, 5.59314981161876502315638963320, 6.06653547310959621978636638368, 7.26044041738974169680212516423, 8.399892326776514295186608795368, 8.652084536908547434382936656098
