| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s − 11-s + 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 4·19-s − 22-s + 4·23-s + 6·26-s − 28-s + 6·29-s + 32-s + 2·34-s − 3·36-s + 2·37-s − 4·38-s − 6·41-s + 4·43-s − 44-s + 4·46-s − 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 0.301·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.213·22-s + 0.834·23-s + 1.17·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.937·41-s + 0.609·43-s − 0.150·44-s + 0.589·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.840862718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.840862718\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.527044098356721696869996324504, −7.79139251965102174785456011339, −6.68527848715894114881498521882, −6.24105337498979840969416095552, −5.54385508956341705884081923791, −4.75563880056112408272227374276, −3.71728152784183348671977379261, −3.16684972416390811064155896405, −2.23600489601986428512099858026, −0.883283840484406346937518621516,
0.883283840484406346937518621516, 2.23600489601986428512099858026, 3.16684972416390811064155896405, 3.71728152784183348671977379261, 4.75563880056112408272227374276, 5.54385508956341705884081923791, 6.24105337498979840969416095552, 6.68527848715894114881498521882, 7.79139251965102174785456011339, 8.527044098356721696869996324504
