| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·9-s − 3·11-s + 2·12-s + 13-s − 4·16-s + 7·17-s + 4·18-s + 6·22-s − 6·23-s − 2·26-s − 5·27-s − 5·29-s − 2·31-s + 8·32-s − 3·33-s − 14·34-s − 4·36-s − 2·37-s + 39-s − 2·41-s + 4·43-s − 6·44-s + 12·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 0.277·13-s − 16-s + 1.69·17-s + 0.942·18-s + 1.27·22-s − 1.25·23-s − 0.392·26-s − 0.962·27-s − 0.928·29-s − 0.359·31-s + 1.41·32-s − 0.522·33-s − 2.40·34-s − 2/3·36-s − 0.328·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.904·44-s + 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−9.312476513372407303797734264888, −8.441462896647731068716323599650, −7.899757105462614146027162172590, −7.41888454651299846322762526113, −6.08144363048696633696684860610, −5.23822087476164467701837988393, −3.77164180563413971220507497899, −2.71946351618484182731760782003, −1.60049913551789401165977051774, 0,
1.60049913551789401165977051774, 2.71946351618484182731760782003, 3.77164180563413971220507497899, 5.23822087476164467701837988393, 6.08144363048696633696684860610, 7.41888454651299846322762526113, 7.899757105462614146027162172590, 8.441462896647731068716323599650, 9.312476513372407303797734264888
