| L(s) = 1 | + (−4.5 + 2.59i)3-s + (−2.25 − 1.30i)5-s + (16.7 + 46.0i)7-s + (13.5 − 23.3i)9-s + (−63.9 − 110. i)11-s + 246. i·13-s + 13.5·15-s + (299. − 172. i)17-s + (−386. − 223. i)19-s + (−194. − 163. i)21-s + (−321. + 556. i)23-s + (−309. − 535. i)25-s + 140. i·27-s − 1.54e3·29-s + (−1.00e3 + 579. i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (−0.0903 − 0.0521i)5-s + (0.340 + 0.940i)7-s + (0.166 − 0.288i)9-s + (−0.528 − 0.915i)11-s + 1.45i·13-s + 0.0602·15-s + (1.03 − 0.598i)17-s + (−1.07 − 0.618i)19-s + (−0.441 − 0.371i)21-s + (−0.607 + 1.05i)23-s + (−0.494 − 0.856i)25-s + 0.192i·27-s − 1.83·29-s + (−1.04 + 0.603i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1784906877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1784906877\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 + (-16.7 - 46.0i)T \) |
| good | 5 | \( 1 + (2.25 + 1.30i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (63.9 + 110. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 246. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-299. + 172. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (386. + 223. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (321. - 556. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 1.54e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (1.00e3 - 579. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-701. + 1.21e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.65e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.73e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (581. + 335. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-1.99e3 - 3.46e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (928. - 536. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (3.98e3 + 2.29e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.84e3 - 3.18e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 1.20e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-3.61e3 + 2.08e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.59e3 - 6.22e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 664. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-1.02e4 - 5.89e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 3.92e3iT - 8.85e7T^{2} \) |
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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
−12.38680456636804538522026610532, −11.59275555867855991954313885720, −10.88567146817983627086907575326, −9.512676919402707154238015539199, −8.731456258116158500169608754966, −7.44119847178058000172513181263, −6.02706587603556892754225730868, −5.20619432188904855038862788518, −3.77686130319824880414433782902, −2.02855222909271010474344586345,
0.06874215599536431937547602171, 1.72183375720038803063645114826, 3.67475526450227800046895511180, 5.02237533543121375458052721618, 6.16540837160511318622420320996, 7.58233846908609883932083425839, 8.024865974010502712425805288593, 9.968299576336697045807259957859, 10.48509530240664313241084049205, 11.52479045289975725224476889909
