| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 11-s − 13-s + 4·14-s + 16-s − 2·17-s − 20-s + 22-s + 25-s − 26-s + 4·28-s − 8·29-s + 8·31-s + 32-s − 2·34-s − 4·35-s − 2·37-s − 40-s + 4·43-s + 44-s + 6·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.213·22-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 1.48·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.676·35-s − 0.328·37-s − 0.158·40-s + 0.609·43-s + 0.150·44-s + 0.875·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.016724334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.016724334\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.14050687156576, −15.47123560056188, −15.07427984214202, −14.67698991812048, −13.87760155542218, −13.75961829028207, −12.80951951959507, −12.24326855042926, −11.74453617461586, −11.19600756329793, −10.85243324502048, −10.09015227867278, −9.223539529666353, −8.592620486301173, −7.944286908265538, −7.466324544102502, −6.830558552492644, −6.003962773192057, −5.315371761658039, −4.692223979936573, −4.206207166082252, −3.495660073308850, −2.453015781160685, −1.836740021998411, −0.8259109728916828,
0.8259109728916828, 1.836740021998411, 2.453015781160685, 3.495660073308850, 4.206207166082252, 4.692223979936573, 5.315371761658039, 6.003962773192057, 6.830558552492644, 7.466324544102502, 7.944286908265538, 8.592620486301173, 9.223539529666353, 10.09015227867278, 10.85243324502048, 11.19600756329793, 11.74453617461586, 12.24326855042926, 12.80951951959507, 13.75961829028207, 13.87760155542218, 14.67698991812048, 15.07427984214202, 15.47123560056188, 16.14050687156576
