| L(s) = 1 | + 5·2-s + 17·4-s − 5·5-s + 9·7-s + 45·8-s − 25·10-s + 8·11-s + 43·13-s + 45·14-s + 89·16-s + 122·17-s − 59·19-s − 85·20-s + 40·22-s + 213·23-s + 25·25-s + 215·26-s + 153·28-s − 224·29-s − 36·31-s + 85·32-s + 610·34-s − 45·35-s + 206·37-s − 295·38-s − 225·40-s − 413·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s − 0.447·5-s + 0.485·7-s + 1.98·8-s − 0.790·10-s + 0.219·11-s + 0.917·13-s + 0.859·14-s + 1.39·16-s + 1.74·17-s − 0.712·19-s − 0.950·20-s + 0.387·22-s + 1.93·23-s + 1/5·25-s + 1.62·26-s + 1.03·28-s − 1.43·29-s − 0.208·31-s + 0.469·32-s + 3.07·34-s − 0.217·35-s + 0.915·37-s − 1.25·38-s − 0.889·40-s − 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.842491084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.842491084\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 122 T + p^{3} T^{2} \) |
| 19 | \( 1 + 59 T + p^{3} T^{2} \) |
| 23 | \( 1 - 213 T + p^{3} T^{2} \) |
| 29 | \( 1 + 224 T + p^{3} T^{2} \) |
| 31 | \( 1 + 36 T + p^{3} T^{2} \) |
| 37 | \( 1 - 206 T + p^{3} T^{2} \) |
| 41 | \( 1 + 413 T + p^{3} T^{2} \) |
| 43 | \( 1 + 392 T + p^{3} T^{2} \) |
| 47 | \( 1 - 311 T + p^{3} T^{2} \) |
| 53 | \( 1 - 377 T + p^{3} T^{2} \) |
| 59 | \( 1 + 337 T + p^{3} T^{2} \) |
| 61 | \( 1 - 40 T + p^{3} T^{2} \) |
| 67 | \( 1 - 348 T + p^{3} T^{2} \) |
| 71 | \( 1 + 62 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1214 T + p^{3} T^{2} \) |
| 79 | \( 1 + 294 T + p^{3} T^{2} \) |
| 83 | \( 1 + 534 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 928 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30962561064121288392848382827, −10.38574502479962415276032027645, −8.861550629330782766802188025029, −7.71229153023281799513618686793, −6.81042693814288051823667141934, −5.73737992324121311528520239993, −4.95655266932177802309775364118, −3.85439490187708442282997125413, −3.09656586138044454814294466786, −1.46678165799877809001805792097,
1.46678165799877809001805792097, 3.09656586138044454814294466786, 3.85439490187708442282997125413, 4.95655266932177802309775364118, 5.73737992324121311528520239993, 6.81042693814288051823667141934, 7.71229153023281799513618686793, 8.861550629330782766802188025029, 10.38574502479962415276032027645, 11.30962561064121288392848382827
