| L(s) = 1 | + (−0.830 + 1.81i)2-s + (2.87 + 0.845i)3-s + (−2.61 − 3.02i)4-s + (8.44 − 5.42i)5-s + (−3.92 + 4.53i)6-s + (2.02 − 14.0i)7-s + (7.67 − 2.25i)8-s + (7.57 + 4.86i)9-s + (2.85 + 19.8i)10-s + (−19.1 − 42.0i)11-s + (−4.98 − 10.9i)12-s + (−4.90 − 34.1i)13-s + (23.9 + 15.3i)14-s + (28.8 − 8.47i)15-s + (−2.27 + 15.8i)16-s + (60.0 − 69.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.754 − 0.485i)5-s + (−0.267 + 0.308i)6-s + (0.109 − 0.759i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.0903 + 0.628i)10-s + (−0.525 − 1.15i)11-s + (−0.119 − 0.262i)12-s + (−0.104 − 0.727i)13-s + (0.456 + 0.293i)14-s + (0.497 − 0.145i)15-s + (−0.0355 + 0.247i)16-s + (0.856 − 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.79834 - 0.221627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79834 - 0.221627i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 - 1.81i)T \) |
| 3 | \( 1 + (-2.87 - 0.845i)T \) |
| 23 | \( 1 + (44.2 - 101. i)T \) |
| good | 5 | \( 1 + (-8.44 + 5.42i)T + (51.9 - 113. i)T^{2} \) |
| 7 | \( 1 + (-2.02 + 14.0i)T + (-329. - 96.6i)T^{2} \) |
| 11 | \( 1 + (19.1 + 42.0i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (4.90 + 34.1i)T + (-2.10e3 + 618. i)T^{2} \) |
| 17 | \( 1 + (-60.0 + 69.2i)T + (-699. - 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-53.9 - 62.2i)T + (-976. + 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-30.5 + 35.2i)T + (-3.47e3 - 2.41e4i)T^{2} \) |
| 31 | \( 1 + (-231. + 68.0i)T + (2.50e4 - 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-255. - 163. i)T + (2.10e4 + 4.60e4i)T^{2} \) |
| 41 | \( 1 + (161. - 103. i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-25.1 - 7.37i)T + (6.68e4 + 4.29e4i)T^{2} \) |
| 47 | \( 1 + 606.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-37.9 + 263. i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (92.0 + 640. i)T + (-1.97e5 + 5.78e4i)T^{2} \) |
| 61 | \( 1 + (483. - 142. i)T + (1.90e5 - 1.22e5i)T^{2} \) |
| 67 | \( 1 + (203. - 445. i)T + (-1.96e5 - 2.27e5i)T^{2} \) |
| 71 | \( 1 + (301. - 660. i)T + (-2.34e5 - 2.70e5i)T^{2} \) |
| 73 | \( 1 + (27.1 + 31.3i)T + (-5.53e4 + 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-6.26 - 43.5i)T + (-4.73e5 + 1.38e5i)T^{2} \) |
| 83 | \( 1 + (86.2 + 55.4i)T + (2.37e5 + 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-400. - 117. i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-152. + 98.3i)T + (3.79e5 - 8.30e5i)T^{2} \) |
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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
−13.25678592926941165402282672777, −11.56680452942805570148888707827, −10.10785763690758050128945305642, −9.655778689328758055000991136187, −8.249017620591631641179101275456, −7.63292663949056140599306883340, −5.99676851548432226666957340306, −5.01342031040883045787270816721, −3.23894236416154711470336753875, −1.03598962504712975811087131547,
1.86062338152424292219834504939, 2.82151719438236135488989760727, 4.63711098151024294194954255437, 6.25443413553886260915831117118, 7.59821464200237625424496919641, 8.758117838675709305993362218914, 9.771180508690051132106636359896, 10.42628644339362998079370746284, 11.89498436066399620529215118859, 12.61416459274994356176826923152
