| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s − 2·11-s − 12-s + 2·13-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s − 2·20-s − 2·22-s − 24-s − 25-s + 2·26-s − 27-s + 2·29-s + 2·30-s − 6·31-s + 32-s + 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.426·22-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s − 1.07·31-s + 0.176·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.94331448791856, −13.36226802214839, −12.96772819047579, −12.66349954826097, −11.93373891851566, −11.52882199054318, −11.25830649968027, −10.85598339917921, −10.05130276546634, −9.822207920570480, −8.986591018637143, −8.321233693580013, −7.978569236136773, −7.297892927761655, −6.982580421124689, −6.347491577980526, −5.711067637045890, −5.335859040571629, −4.709468861749431, −4.219160445207260, −3.605985534960107, −3.188955365595545, −2.383895069112183, −1.650765187911195, −0.8170283080410596, 0,
0.8170283080410596, 1.650765187911195, 2.383895069112183, 3.188955365595545, 3.605985534960107, 4.219160445207260, 4.709468861749431, 5.335859040571629, 5.711067637045890, 6.347491577980526, 6.982580421124689, 7.297892927761655, 7.978569236136773, 8.321233693580013, 8.986591018637143, 9.822207920570480, 10.05130276546634, 10.85598339917921, 11.25830649968027, 11.52882199054318, 11.93373891851566, 12.66349954826097, 12.96772819047579, 13.36226802214839, 13.94331448791856
