| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 3·11-s + 12-s + 2·13-s + 14-s + 16-s + 3·17-s − 2·18-s − 7·19-s + 21-s + 3·22-s + 24-s + 2·26-s − 5·27-s + 28-s − 6·29-s − 4·31-s + 32-s + 3·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 1.60·19-s + 0.218·21-s + 0.639·22-s + 0.204·24-s + 0.392·26-s − 0.962·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.403179303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.403179303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−11.45909715795481371039597944171, −10.89274588760446240115568394507, −9.536378818315516005204945079997, −8.610022790516695549795853061214, −7.77658810173983217795932841222, −6.50150023662226984648693008298, −5.63302791327341414299631153495, −4.26850617197803948832200761616, −3.32408442847114247075118763103, −1.89984331795161626562592295843,
1.89984331795161626562592295843, 3.32408442847114247075118763103, 4.26850617197803948832200761616, 5.63302791327341414299631153495, 6.50150023662226984648693008298, 7.77658810173983217795932841222, 8.610022790516695549795853061214, 9.536378818315516005204945079997, 10.89274588760446240115568394507, 11.45909715795481371039597944171
