| L(s) = 1 | + (−0.724 + 1.25i)2-s + (−4.59 + 2.42i)3-s + (2.95 + 5.11i)4-s + 4.43·5-s + (0.283 − 7.51i)6-s + (−18.5 + 0.531i)7-s − 20.1·8-s + (15.2 − 22.2i)9-s + (−3.20 + 5.55i)10-s − 52.4·11-s + (−25.9 − 16.3i)12-s + (−21.3 + 36.9i)13-s + (12.7 − 23.6i)14-s + (−20.3 + 10.7i)15-s + (−9.03 + 15.6i)16-s + (54.8 − 94.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.255 + 0.443i)2-s + (−0.884 + 0.466i)3-s + (0.368 + 0.639i)4-s + 0.396·5-s + (0.0193 − 0.511i)6-s + (−0.999 + 0.0287i)7-s − 0.889·8-s + (0.563 − 0.825i)9-s + (−0.101 + 0.175i)10-s − 1.43·11-s + (−0.624 − 0.392i)12-s + (−0.455 + 0.788i)13-s + (0.243 − 0.450i)14-s + (−0.350 + 0.185i)15-s + (−0.141 + 0.244i)16-s + (0.782 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0643i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0177713 - 0.551895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0177713 - 0.551895i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.59 - 2.42i)T \) |
| 7 | \( 1 + (18.5 - 0.531i)T \) |
| good | 2 | \( 1 + (0.724 - 1.25i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 4.43T + 125T^{2} \) |
| 11 | \( 1 + 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + (21.3 - 36.9i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-54.8 + 94.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-72.6 - 125. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 37.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-97.0 - 168. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-101. - 175. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-81.1 - 140. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (52.5 - 90.9i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (103. + 179. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (37.5 - 64.9i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (213. - 369. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (73.8 + 127. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-130. + 226. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (219. + 380. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 356.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-190. + 329. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (10.9 - 18.9i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-525. - 910. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (288. + 500. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-564. - 978. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.62507100336319502535798164840, −13.93346625403167332565952937721, −12.48407375054179928318693026480, −11.86861433269779989170446518003, −10.26544504472269394801082696070, −9.474585262636818984980426627430, −7.68493991390817775418572445910, −6.51441628486505270558307241335, −5.29544739436185174203526108669, −3.18468299928532091948050014566,
0.42745430845814127409767402800, 2.53607163949702149020724115462, 5.40785060514713731463930274582, 6.24122476402362610190084941057, 7.72107016035692468543939985709, 9.877536989540429703734628636235, 10.31110288988549428865264818262, 11.54930260218226715896050348545, 12.72788642128897498562448327799, 13.48898325920369849122679822599
