| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 6·11-s + 12-s − 2·13-s + 16-s + 17-s + 18-s + 4·19-s + 6·22-s − 5·23-s + 24-s − 2·26-s + 27-s − 2·31-s + 32-s + 6·33-s + 34-s + 36-s + 3·37-s + 4·38-s − 2·39-s + 5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 1.27·22-s − 1.04·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.359·31-s + 0.176·32-s + 1.04·33-s + 0.171·34-s + 1/6·36-s + 0.493·37-s + 0.648·38-s − 0.320·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.990048983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.990048983\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−9.025513644312571772684716624187, −7.982599482909319769142017170298, −7.34291924238588986650872316517, −6.53703348803029039908614729311, −5.84480516405545554790906042027, −4.79534774139369369545108486670, −3.98707799932357704297545954199, −3.36841003026765223865972449073, −2.26852252619174190811479308711, −1.23630003518010917776962051379,
1.23630003518010917776962051379, 2.26852252619174190811479308711, 3.36841003026765223865972449073, 3.98707799932357704297545954199, 4.79534774139369369545108486670, 5.84480516405545554790906042027, 6.53703348803029039908614729311, 7.34291924238588986650872316517, 7.982599482909319769142017170298, 9.025513644312571772684716624187
