| L(s) = 1 | − 2-s + 4-s − 4·5-s − 7-s − 8-s + 4·10-s + 2·13-s + 14-s + 16-s − 17-s − 19-s − 4·20-s + 6·23-s + 11·25-s − 2·26-s − 28-s + 2·29-s − 4·31-s − 32-s + 34-s + 4·35-s + 2·37-s + 38-s + 4·40-s + 8·41-s + 4·43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s + 1.26·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.229·19-s − 0.894·20-s + 1.25·23-s + 11/5·25-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.676·35-s + 0.328·37-s + 0.162·38-s + 0.632·40-s + 1.24·41-s + 0.609·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.94425674446139, −12.49648743955782, −12.07493247195007, −11.48079691685532, −11.21919938465830, −10.82211103386134, −10.44046222712104, −9.753180334613775, −9.142221779991294, −8.867839730145140, −8.356116713152194, −8.001074292884145, −7.323872607002662, −7.173565200476346, −6.658571347119276, −6.022534991238157, −5.467647789910161, −4.747667455706998, −4.249236886295495, −3.743147603188715, −3.325185679992931, −2.701090042202942, −2.152881950206121, −1.020595779209039, −0.7941011163563296, 0,
0.7941011163563296, 1.020595779209039, 2.152881950206121, 2.701090042202942, 3.325185679992931, 3.743147603188715, 4.249236886295495, 4.747667455706998, 5.467647789910161, 6.022534991238157, 6.658571347119276, 7.173565200476346, 7.323872607002662, 8.001074292884145, 8.356116713152194, 8.867839730145140, 9.142221779991294, 9.753180334613775, 10.44046222712104, 10.82211103386134, 11.21919938465830, 11.48079691685532, 12.07493247195007, 12.49648743955782, 12.94425674446139
