| L(s) = 1 | + (2.81 − 19.5i)3-s + (−3.30 + 0.969i)5-s + (157. + 182. i)7-s + (−140. − 41.3i)9-s + (597. + 383. i)11-s + (200. − 231. i)13-s + (9.66 + 67.2i)15-s + (−525. − 1.15e3i)17-s + (258. − 566. i)19-s + (4.00e3 − 2.57e3i)21-s + (−1.13e3 − 2.26e3i)23-s + (−2.61e3 + 1.68e3i)25-s + (787. − 1.72e3i)27-s + (2.65e3 + 5.81e3i)29-s + (−99.0 − 689. i)31-s + ⋯ |
| L(s) = 1 | + (0.180 − 1.25i)3-s + (−0.0590 + 0.0173i)5-s + (1.21 + 1.40i)7-s + (−0.580 − 0.170i)9-s + (1.48 + 0.956i)11-s + (0.329 − 0.380i)13-s + (0.0110 + 0.0771i)15-s + (−0.441 − 0.965i)17-s + (0.164 − 0.359i)19-s + (1.98 − 1.27i)21-s + (−0.447 − 0.894i)23-s + (−0.838 + 0.538i)25-s + (0.208 − 0.455i)27-s + (0.586 + 1.28i)29-s + (−0.0185 − 0.128i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.25694 - 0.668535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25694 - 0.668535i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (1.13e3 + 2.26e3i)T \) |
| good | 3 | \( 1 + (-2.81 + 19.5i)T + (-233. - 68.4i)T^{2} \) |
| 5 | \( 1 + (3.30 - 0.969i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (-157. - 182. i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (-597. - 383. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (-200. + 231. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (525. + 1.15e3i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (-258. + 566. i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (-2.65e3 - 5.81e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (99.0 + 689. i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (-9.28e3 - 2.72e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (-1.18e4 + 3.47e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (930. - 6.47e3i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 - 5.91e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.15e4 + 1.33e4i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (-1.54e4 + 1.78e4i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (5.50e3 + 3.82e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (5.21e4 - 3.35e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (-6.65e3 + 4.27e3i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (1.58e4 - 3.48e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (-5.12e4 + 5.91e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (9.82e4 + 2.88e4i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (7.17e3 - 4.99e4i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (-6.86e4 + 2.01e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−12.80190387716949960550241457448, −12.00862799936140758373173690569, −11.34914400074109874155163020367, −9.374229284133246619910258200571, −8.418507368370636466371848227032, −7.33746242351037732437028080593, −6.20394543570838435424452415850, −4.69386064709648188400141237228, −2.38886925339013823407640842905, −1.34181176413204057913405886102,
1.26011701122749270366942029523, 3.98969226768843535453262030415, 4.18801163280361463998808555444, 6.09289445384725342438826892262, 7.75365813262999552107986897117, 8.871056347733284365767048178963, 10.02695522879747183935928471401, 10.97751884800803171604559415964, 11.68560201540598642132481765855, 13.67863512732734917222174933041
