| L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s + 3·8-s + 9-s + 2·10-s − 12-s − 2·13-s − 2·15-s − 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s + 8·23-s + 3·24-s − 25-s + 2·26-s + 27-s − 6·29-s + 2·30-s − 4·31-s − 5·32-s − 2·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.632·10-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 + 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.365·30-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048334003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048334003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.918617285487690578222083522732, −7.56501130434046089275199924789, −7.24071715552599050064456361648, −5.93069296846352306452428795644, −5.01781465592685407991604577699, −4.39653187738690261424623244258, −3.59379316009426175283933742018, −2.88462717277435777791911028004, −1.61059515248252936637538351208, −0.61113971392744732136662819427,
0.61113971392744732136662819427, 1.61059515248252936637538351208, 2.88462717277435777791911028004, 3.59379316009426175283933742018, 4.39653187738690261424623244258, 5.01781465592685407991604577699, 5.93069296846352306452428795644, 7.24071715552599050064456361648, 7.56501130434046089275199924789, 7.918617285487690578222083522732
