| L(s) = 1 | + 3.86i·2-s + 1.07·4-s − 17.4i·5-s + 88.9·7-s + 65.9i·8-s + 67.3·10-s − 118. i·11-s + 299.·13-s + 343. i·14-s − 237.·16-s + 163. i·17-s − 57.9·19-s − 18.7i·20-s + 457.·22-s − 461. i·23-s + ⋯ |
| L(s) = 1 | + 0.965i·2-s + 0.0673·4-s − 0.697i·5-s + 1.81·7-s + 1.03i·8-s + 0.673·10-s − 0.978i·11-s + 1.77·13-s + 1.75i·14-s − 0.928·16-s + 0.565i·17-s − 0.160·19-s − 0.0469i·20-s + 0.945·22-s − 0.873i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.366159672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.366159672\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 453. iT \) |
| good | 2 | \( 1 - 3.86iT - 16T^{2} \) |
| 5 | \( 1 + 17.4iT - 625T^{2} \) |
| 7 | \( 1 - 88.9T + 2.40e3T^{2} \) |
| 11 | \( 1 + 118. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 299.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 163. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 57.9T + 1.30e5T^{2} \) |
| 23 | \( 1 + 461. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 897. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.16e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.01e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 365. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.35e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.49e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.59e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 - 2.77e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.01e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.95e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.87e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.89e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.59e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.05e3T + 8.85e7T^{2} \) |
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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
−10.62346276196748448231521855882, −8.715168366500790080016173632173, −8.486800094470726076363927948683, −7.87990086189574085771687679513, −6.58287103970329184069390926356, −5.71651155198932737112214364864, −4.99381535586970026621283958763, −3.86077756236451011135108555314, −2.03024906551357052154609608885, −0.990924878278591166624863281572,
1.27323386756035128761818356000, 1.87260978120948206017525253320, 3.16312680435485326152439012451, 4.19298368763661310777471830462, 5.29636381525895201629001195955, 6.64820535239937544071031591408, 7.44339786212591119913848545956, 8.425253689977357241920343740110, 9.485364574936297837242016318452, 10.54908032297185851699693927661
