| L(s) = 1 | + (−1.80 − 0.867i)2-s + (0.477 − 0.598i)3-s + (2.49 + 3.12i)4-s + (1.70 + 0.820i)5-s + (−1.37 + 0.664i)6-s + (16.1 − 20.2i)7-s + (−1.78 − 7.79i)8-s + (5.87 + 25.7i)9-s + (−2.35 − 2.95i)10-s + (14.8 − 65.2i)11-s + 3.06·12-s + (10.9 − 47.8i)13-s + (−46.6 + 22.4i)14-s + (1.30 − 0.627i)15-s + (−3.56 + 15.5i)16-s + 26.2·17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (0.0918 − 0.115i)3-s + (0.311 + 0.390i)4-s + (0.152 + 0.0733i)5-s + (−0.0938 + 0.0451i)6-s + (0.871 − 1.09i)7-s + (−0.0786 − 0.344i)8-s + (0.217 + 0.953i)9-s + (−0.0745 − 0.0935i)10-s + (0.408 − 1.78i)11-s + 0.0736·12-s + (0.232 − 1.02i)13-s + (−0.890 + 0.428i)14-s + (0.0224 − 0.0108i)15-s + (−0.0556 + 0.243i)16-s + 0.374·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.06985 - 0.548189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06985 - 0.548189i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.80 + 0.867i)T \) |
| 29 | \( 1 + (56.3 - 145. i)T \) |
| good | 3 | \( 1 + (-0.477 + 0.598i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-1.70 - 0.820i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (-16.1 + 20.2i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (-14.8 + 65.2i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-10.9 + 47.8i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 - 26.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-94.6 - 118. i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (171. - 82.6i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (-134. - 64.6i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 6.31i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-238. + 114. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-29.0 + 127. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (122. + 58.8i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + 573.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (346. - 434. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-200. - 876. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-89.3 + 391. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-143. + 69.2i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (-90.9 - 398. i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (-492. - 616. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-716. - 344. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (-168. - 211. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−14.04270613038203749433398632841, −13.73002762668694075036283073303, −11.95826370706700703619761853074, −10.82447549298918222812379248110, −10.12635661568367234123415309861, −8.233933242474738289828513832770, −7.71231750645364022188515326366, −5.71086621807635432729985630277, −3.59051816329173504476227706665, −1.26277308289461992248708926545,
1.92288449272385199894484726167, 4.62912417255800135355038093261, 6.31051562602917322033119572634, 7.66040473919939773246296760764, 9.177504985629929834617511802599, 9.683241781063548438724480171258, 11.59479009404384384012195662774, 12.19944502518691588665713153422, 14.07593521524945760584611461662, 15.13051009431166345180776682923
