| L(s) = 1 | + 22·2-s + 356·4-s + 125·5-s − 420·7-s + 5.01e3·8-s + 2.75e3·10-s + 2.94e3·11-s − 1.10e4·13-s − 9.24e3·14-s + 6.47e4·16-s + 1.65e4·17-s − 2.53e4·19-s + 4.45e4·20-s + 6.47e4·22-s + 5.88e3·23-s + 1.56e4·25-s − 2.42e5·26-s − 1.49e5·28-s − 1.63e5·29-s − 2.01e5·31-s + 7.83e5·32-s + 3.64e5·34-s − 5.25e4·35-s + 1.20e5·37-s − 5.58e5·38-s + 6.27e5·40-s + 1.15e5·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.78·4-s + 0.447·5-s − 0.462·7-s + 3.46·8-s + 0.869·10-s + 0.666·11-s − 1.38·13-s − 0.899·14-s + 3.95·16-s + 0.816·17-s − 0.848·19-s + 1.24·20-s + 1.29·22-s + 0.100·23-s + 1/5·25-s − 2.70·26-s − 1.28·28-s − 1.24·29-s − 1.21·31-s + 4.22·32-s + 1.58·34-s − 0.206·35-s + 0.391·37-s − 1.64·38-s + 1.54·40-s + 0.262·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.978089246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.978089246\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 - 11 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 60 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 2944 T + p^{7} T^{2} \) |
| 13 | \( 1 + 11006 T + p^{7} T^{2} \) |
| 17 | \( 1 - 16546 T + p^{7} T^{2} \) |
| 19 | \( 1 + 25364 T + p^{7} T^{2} \) |
| 23 | \( 1 - 5880 T + p^{7} T^{2} \) |
| 29 | \( 1 + 163042 T + p^{7} T^{2} \) |
| 31 | \( 1 + 201600 T + p^{7} T^{2} \) |
| 37 | \( 1 - 120530 T + p^{7} T^{2} \) |
| 41 | \( 1 - 115910 T + p^{7} T^{2} \) |
| 43 | \( 1 + 19148 T + p^{7} T^{2} \) |
| 47 | \( 1 + 841016 T + p^{7} T^{2} \) |
| 53 | \( 1 + 501890 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1586176 T + p^{7} T^{2} \) |
| 61 | \( 1 + 372962 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4561044 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1512832 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1522910 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4231920 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1854204 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6888174 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3700034 T + p^{7} T^{2} \) |
| 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
−14.44978273376319662539027933222, −13.08704114095014083100055736729, −12.42510284424430222877950530449, −11.24533865751051091497724771473, −9.828793523850557136254307972011, −7.35438233457823198527120495680, −6.20298061821349618068947673694, −5.00145464621662400621284126901, −3.54536941179123188100747666530, −2.07041480942734506608398712204,
2.07041480942734506608398712204, 3.54536941179123188100747666530, 5.00145464621662400621284126901, 6.20298061821349618068947673694, 7.35438233457823198527120495680, 9.828793523850557136254307972011, 11.24533865751051091497724771473, 12.42510284424430222877950530449, 13.08704114095014083100055736729, 14.44978273376319662539027933222
