| L(s) = 1 | + (1.26 − 0.642i)2-s + (−0.742 + 0.117i)3-s + (1.17 − 1.61i)4-s + (4.02 − 2.96i)5-s + (−0.860 + 0.624i)6-s + (2.71 + 2.71i)7-s + (0.442 − 2.79i)8-s + (−8.02 + 2.60i)9-s + (3.17 − 6.31i)10-s + (−4.47 + 13.7i)11-s + (−0.682 + 1.33i)12-s + (−16.6 − 8.47i)13-s + (5.16 + 1.67i)14-s + (−2.64 + 2.67i)15-s + (−1.23 − 3.80i)16-s + (10.5 + 1.67i)17-s + ⋯ |
| L(s) = 1 | + (0.630 − 0.321i)2-s + (−0.247 + 0.0392i)3-s + (0.293 − 0.404i)4-s + (0.805 − 0.592i)5-s + (−0.143 + 0.104i)6-s + (0.387 + 0.387i)7-s + (0.0553 − 0.349i)8-s + (−0.891 + 0.289i)9-s + (0.317 − 0.631i)10-s + (−0.407 + 1.25i)11-s + (−0.0568 + 0.111i)12-s + (−1.27 − 0.651i)13-s + (0.368 + 0.119i)14-s + (−0.176 + 0.178i)15-s + (−0.0772 − 0.237i)16-s + (0.621 + 0.0985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46158 - 0.391086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46158 - 0.391086i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 0.642i)T \) |
| 5 | \( 1 + (-4.02 + 2.96i)T \) |
| good | 3 | \( 1 + (0.742 - 0.117i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (-2.71 - 2.71i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.47 - 13.7i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (16.6 + 8.47i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-10.5 - 1.67i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-11.3 - 15.6i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (8.15 + 16.0i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (15.1 - 20.8i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-6.46 + 4.70i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-31.9 + 62.7i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.20 - 3.72i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-9.83 + 9.83i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-5.34 - 33.7i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (3.64 - 0.577i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (111. - 36.1i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-9.28 + 28.5i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (62.2 + 9.85i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-94.8 - 68.9i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-35.3 - 69.4i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-85.0 + 117. i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (1.65 - 10.4i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-3.88 - 1.26i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-16.6 - 105. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−14.86456839748608016073677856574, −14.18279622754257466310480396196, −12.66994882273386554135947952870, −12.14654930166234287723536381246, −10.54147327268659612163524724175, −9.505841731358768918097892300394, −7.76193132654301995762494124935, −5.72910729738495783098017550237, −4.91458695480956170201886294777, −2.33443674933814887616750232085,
2.95117800749499345894057129686, 5.18844974575191828410456765069, 6.31788778699386921204626876328, 7.74607630482267146740683698985, 9.486904292864326636913712892153, 11.00354792354778925759356915151, 11.88144515344264745945478069435, 13.59461765689561688549112788270, 14.09653431955126013660283904843, 15.15103135464416277998075468856
