| L(s) = 1 | − 4.81·2-s + 7.21·4-s + 11.1i·5-s + (44.8 + 19.6i)7-s + 42.3·8-s − 53.8i·10-s − 201.·11-s − 258. i·13-s + (−216. − 94.5i)14-s − 319.·16-s + 242. i·17-s + 253. i·19-s + 80.6i·20-s + 971.·22-s + 323.·23-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.450·4-s + 0.447i·5-s + (0.916 + 0.400i)7-s + 0.661·8-s − 0.538i·10-s − 1.66·11-s − 1.53i·13-s + (−1.10 − 0.482i)14-s − 1.24·16-s + 0.840i·17-s + 0.702i·19-s + 0.201i·20-s + 2.00·22-s + 0.611·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3300326922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3300326922\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 11.1iT \) |
| 7 | \( 1 + (-44.8 - 19.6i)T \) |
| good | 2 | \( 1 + 4.81T + 16T^{2} \) |
| 11 | \( 1 + 201.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 258. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 242. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 253. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 323.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 331.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 278. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 322.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 578. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.52e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 389. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.07e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.03e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.68e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.05e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.21e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 2.23e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.23e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.79e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.36e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.85e4iT - 8.85e7T^{2} \) |
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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
−10.51610580767627678146576097179, −10.01325177736646107227201938070, −8.550531291437173431327347418705, −8.081095162289623991844581092879, −7.38152292458815892454065105826, −5.76029624005440861135680416683, −4.82801655431818895863692082892, −3.00524910201109418739069264785, −1.70015030985860450185350253613, −0.17214833683557155094410176309,
1.11876956778067430116684424521, 2.37299317970280262369136743145, 4.47469569731395990262688778395, 5.11578980855181586404854037318, 6.96721713419382361344352212164, 7.70199616939361448260441079980, 8.568322925199897631915257857691, 9.305280137983979747154195224528, 10.29477554805920949520050072937, 11.07653700868255550373001397398
