| L(s) = 1 | + (−1.19 + 4.45i)2-s + (−12.4 − 3.33i)3-s + (−4.59 − 2.65i)4-s + (16.0 + 19.2i)5-s + (29.7 − 51.4i)6-s + (56.2 + 56.2i)7-s + (−34.8 + 34.8i)8-s + (73.5 + 42.4i)9-s + (−104. + 48.4i)10-s − 0.443·11-s + (48.3 + 48.3i)12-s + (7.53 + 28.1i)13-s + (−317. + 183. i)14-s + (−135. − 292. i)15-s + (−156. − 270. i)16-s + (5.27 − 19.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.298 + 1.11i)2-s + (−1.38 − 0.370i)3-s + (−0.287 − 0.165i)4-s + (0.640 + 0.768i)5-s + (0.825 − 1.43i)6-s + (1.14 + 1.14i)7-s + (−0.545 + 0.545i)8-s + (0.907 + 0.524i)9-s + (−1.04 + 0.484i)10-s − 0.00366·11-s + (0.335 + 0.335i)12-s + (0.0445 + 0.166i)13-s + (−1.62 + 0.936i)14-s + (−0.600 − 1.29i)15-s + (−0.610 − 1.05i)16-s + (0.0182 − 0.0681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.162794 - 0.815329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162794 - 0.815329i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-16.0 - 19.2i)T \) |
| 19 | \( 1 + (358. + 42.4i)T \) |
| good | 2 | \( 1 + (1.19 - 4.45i)T + (-13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (12.4 + 3.33i)T + (70.1 + 40.5i)T^{2} \) |
| 7 | \( 1 + (-56.2 - 56.2i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 0.443T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-7.53 - 28.1i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-5.27 + 19.7i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 23 | \( 1 + (-96.0 - 358. i)T + (-2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 + (1.07e3 + 623. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 - 490.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.54e3 + 1.54e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + (-226. - 391. i)T + (-1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-2.48e3 - 665. i)T + (2.96e6 + 1.70e6i)T^{2} \) |
| 47 | \( 1 + (4.26e3 - 1.14e3i)T + (4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (-538. - 2.00e3i)T + (-6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (175. - 101. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.96e3 + 5.13e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-6.31e3 + 1.69e3i)T + (1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-2.30e3 - 4.00e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-1.02e3 - 274. i)T + (2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (191. - 110. i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (5.08e3 - 5.08e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-4.57e3 - 2.64e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.23e3 - 4.60e3i)T + (-7.66e7 - 4.42e7i)T^{2} \) |
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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
−14.24601451709440201223883646351, −12.63915310071202286942040108854, −11.47927881949073611311227931876, −11.04050510038936597064611161806, −9.289686179873701438313384910989, −7.973605966560872622597023225982, −6.78899731423003109641255608028, −5.90476771382431823909260334767, −5.22664779558964283072656232242, −2.10828528991019293812312230091,
0.50884523779386991678140326801, 1.66815376196526671203288062574, 4.21016523272370585746380290280, 5.28282567303787363038412941603, 6.64961278638049538236349155155, 8.519113795700391602500512842413, 10.00324128020248847827929591701, 10.64988391968940054294893255247, 11.33140143960331238478018651846, 12.31869979604185184698710849055
