| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 3·11-s + 12-s + 5·13-s + 14-s + 16-s − 3·17-s − 18-s + 8·19-s − 21-s − 3·22-s − 6·23-s − 24-s − 5·25-s − 5·26-s + 27-s − 28-s + 6·29-s + 8·31-s − 32-s + 3·33-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 + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.83·19-s − 0.218·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s − 25-s − 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245036556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245036556\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
| 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
−11.54700911945520942748258035226, −10.24849536889226805234188917057, −9.587845767160858639497290253924, −8.687489326741442009138757382649, −7.956229369564304686123325694982, −6.77518295329015749347746155322, −5.96230889473484958243059644955, −4.15200591971816077967275368942, −3.03923494480383630496608724904, −1.39113691119196206386099889711,
1.39113691119196206386099889711, 3.03923494480383630496608724904, 4.15200591971816077967275368942, 5.96230889473484958243059644955, 6.77518295329015749347746155322, 7.956229369564304686123325694982, 8.687489326741442009138757382649, 9.587845767160858639497290253924, 10.24849536889226805234188917057, 11.54700911945520942748258035226
