| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s − 2·9-s + 3·10-s − 12-s + 4·13-s − 14-s + 3·15-s + 16-s + 3·17-s + 2·18-s − 3·20-s − 21-s − 6·23-s + 24-s + 4·25-s − 4·26-s + 5·27-s + 28-s + 3·29-s − 3·30-s − 5·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.670·20-s − 0.218·21-s − 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.784·26-s + 0.962·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| 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
−13.50931867618010, −12.81876724787123, −12.22363623282887, −12.02651736723072, −11.52065914027140, −11.20994239588022, −10.81559118091751, −10.30178729734987, −9.883485628288832, −8.975664728136031, −8.800600771393312, −8.208210900498402, −7.872126946292698, −7.492500643797555, −6.882439262488196, −6.229876766647896, −5.886698670463771, −5.388466074833934, −4.637539013419234, −4.174918699500065, −3.498774053566996, −3.184246450006026, −2.409413815393051, −1.501899812673532, −1.094816703525502, 0, 0,
1.094816703525502, 1.501899812673532, 2.409413815393051, 3.184246450006026, 3.498774053566996, 4.174918699500065, 4.637539013419234, 5.388466074833934, 5.886698670463771, 6.229876766647896, 6.882439262488196, 7.492500643797555, 7.872126946292698, 8.208210900498402, 8.800600771393312, 8.975664728136031, 9.883485628288832, 10.30178729734987, 10.81559118091751, 11.20994239588022, 11.52065914027140, 12.02651736723072, 12.22363623282887, 12.81876724787123, 13.50931867618010
