| L(s) = 1 | + (1.24 − 1.56i)2-s + (0.975 − 4.27i)3-s + (−0.890 − 3.89i)4-s + (−2.22 + 2.78i)5-s + (−5.46 − 6.85i)6-s + (5.92 − 25.9i)7-s + (−7.20 − 3.47i)8-s + (7.00 + 3.37i)9-s + (1.58 + 6.95i)10-s + (−59.9 + 28.8i)11-s − 17.5·12-s + (47.8 − 23.0i)13-s + (−33.2 − 41.6i)14-s + (9.75 + 12.2i)15-s + (−14.4 + 6.94i)16-s + 127.·17-s + ⋯ |
| L(s) = 1 | + (0.440 − 0.552i)2-s + (0.187 − 0.822i)3-s + (−0.111 − 0.487i)4-s + (−0.198 + 0.249i)5-s + (−0.372 − 0.466i)6-s + (0.319 − 1.40i)7-s + (−0.318 − 0.153i)8-s + (0.259 + 0.124i)9-s + (0.0502 + 0.220i)10-s + (−1.64 + 0.791i)11-s − 0.421·12-s + (1.02 − 0.491i)13-s + (−0.633 − 0.794i)14-s + (0.167 + 0.210i)15-s + (−0.225 + 0.108i)16-s + 1.81·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.05920 - 1.42548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05920 - 1.42548i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 1.56i)T \) |
| 29 | \( 1 + (-2.97 - 156. i)T \) |
| good | 3 | \( 1 + (-0.975 + 4.27i)T + (-24.3 - 11.7i)T^{2} \) |
| 5 | \( 1 + (2.22 - 2.78i)T + (-27.8 - 121. i)T^{2} \) |
| 7 | \( 1 + (-5.92 + 25.9i)T + (-309. - 148. i)T^{2} \) |
| 11 | \( 1 + (59.9 - 28.8i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-47.8 + 23.0i)T + (1.36e3 - 1.71e3i)T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-5.73 - 25.1i)T + (-6.17e3 + 2.97e3i)T^{2} \) |
| 23 | \( 1 + (-31.4 - 39.4i)T + (-2.70e3 + 1.18e4i)T^{2} \) |
| 31 | \( 1 + (2.49 - 3.12i)T + (-6.62e3 - 2.90e4i)T^{2} \) |
| 37 | \( 1 + (205. + 98.8i)T + (3.15e4 + 3.96e4i)T^{2} \) |
| 41 | \( 1 + 87.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + (96.7 + 121. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (132. - 63.7i)T + (6.47e4 - 8.11e4i)T^{2} \) |
| 53 | \( 1 + (-255. + 320. i)T + (-3.31e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + 613.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (122. - 537. i)T + (-2.04e5 - 9.84e4i)T^{2} \) |
| 67 | \( 1 + (268. + 129. i)T + (1.87e5 + 2.35e5i)T^{2} \) |
| 71 | \( 1 + (-231. + 111. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (453. + 568. i)T + (-8.65e4 + 3.79e5i)T^{2} \) |
| 79 | \( 1 + (-733. - 353. i)T + (3.07e5 + 3.85e5i)T^{2} \) |
| 83 | \( 1 + (-221. - 971. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (801. - 1.00e3i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + (-62.7 - 274. i)T + (-8.22e5 + 3.95e5i)T^{2} \) |
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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
−13.93578600509549504128218698811, −13.21725729670885770155839672803, −12.37837042808302173416668404787, −10.75534916215669623584108168283, −10.20305169002854226953113722458, −7.898894858762933888625249162906, −7.21188925667482494823447214134, −5.20329689933083015594724828147, −3.42688360246523658525624054755, −1.33120448849029498410506129169,
3.15347426354984022738691028321, 4.88888799899998856838008991650, 5.92741562544104890443703234466, 8.010731129370079792395480532173, 8.862454668719584627373803538221, 10.31641974298432351434371496719, 11.75871442412836784742238405733, 12.84056213471259319905492179241, 14.10684560949742489525810078171, 15.32692510844469295142652298902
