| L(s) = 1 | + (−1.59 − 2.76i)2-s + (−0.340 − 8.99i)3-s + (2.90 − 5.03i)4-s + (24.0 − 6.69i)5-s + (−24.3 + 15.2i)6-s + (21.0 − 12.1i)7-s − 69.6·8-s + (−80.7 + 6.11i)9-s + (−56.9 − 55.8i)10-s + (−35.8 + 20.6i)11-s + (−46.2 − 24.4i)12-s + (78.1 + 45.1i)13-s + (−67.3 − 38.8i)14-s + (−68.4 − 214. i)15-s + (64.6 + 111. i)16-s − 318.·17-s + ⋯ |
| L(s) = 1 | + (−0.399 − 0.691i)2-s + (−0.0377 − 0.999i)3-s + (0.181 − 0.314i)4-s + (0.963 − 0.267i)5-s + (−0.675 + 0.424i)6-s + (0.430 − 0.248i)7-s − 1.08·8-s + (−0.997 + 0.0755i)9-s + (−0.569 − 0.558i)10-s + (−0.295 + 0.170i)11-s + (−0.321 − 0.169i)12-s + (0.462 + 0.267i)13-s + (−0.343 − 0.198i)14-s + (−0.304 − 0.952i)15-s + (0.252 + 0.437i)16-s − 1.10·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.357044 - 1.33855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.357044 - 1.33855i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.340 + 8.99i)T \) |
| 5 | \( 1 + (-24.0 + 6.69i)T \) |
| good | 2 | \( 1 + (1.59 + 2.76i)T + (-8 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-21.0 + 12.1i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (35.8 - 20.6i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-78.1 - 45.1i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 318.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 572.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (93.6 - 162. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-1.20e3 + 694. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-91.5 + 158. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 348. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.88e3 + 1.09e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.19e3 + 689. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (342. + 592. i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 2.09e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-4.76e3 - 2.75e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (3.22e3 + 5.58e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-4.14e3 - 2.39e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 6.78e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.04e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (2.10e3 + 3.64e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-3.10e3 - 5.38e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.91e3 + 1.10e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−14.17857732567338264531897465662, −13.43469462678025689774791486382, −12.07933028990746222001911800481, −11.08699114341516445754039055797, −9.805020417645161935731548154239, −8.551312374155991632919826845365, −6.79134053708876873092607129462, −5.51086189493839844677915371648, −2.38709898371292662630780985567, −1.08683524856552405356876254649,
2.96472158513931674503388237329, 5.22555066072281941995021120929, 6.52461014239753818735212746296, 8.303056493863825469025340314285, 9.306880507589168290888540747635, 10.58659274469195107203705938020, 11.80332863724132366690276271061, 13.56493981796005330751278688902, 14.73338796411824378209083354999, 15.75569029487280462411036961985
