| L(s) = 1 | + (−1.59 + 0.919i)2-s + (−5.10 − 0.994i)3-s + (−2.31 + 4.00i)4-s + 0.414·5-s + (9.03 − 3.10i)6-s + (12.7 − 13.4i)7-s − 23.1i·8-s + (25.0 + 10.1i)9-s + (−0.659 + 0.381i)10-s − 49.6i·11-s + (15.7 − 18.1i)12-s + (1.43 − 0.828i)13-s + (−7.94 + 33.1i)14-s + (−2.11 − 0.412i)15-s + (2.83 + 4.91i)16-s + (−20.6 − 35.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.562 + 0.324i)2-s + (−0.981 − 0.191i)3-s + (−0.288 + 0.500i)4-s + 0.0370·5-s + (0.614 − 0.211i)6-s + (0.688 − 0.725i)7-s − 1.02i·8-s + (0.926 + 0.375i)9-s + (−0.0208 + 0.0120i)10-s − 1.35i·11-s + (0.379 − 0.435i)12-s + (0.0306 − 0.0176i)13-s + (−0.151 + 0.631i)14-s + (−0.0363 − 0.00709i)15-s + (0.0443 + 0.0767i)16-s + (−0.294 − 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.582434 - 0.289303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582434 - 0.289303i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.10 + 0.994i)T \) |
| 7 | \( 1 + (-12.7 + 13.4i)T \) |
| good | 2 | \( 1 + (1.59 - 0.919i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 0.414T + 125T^{2} \) |
| 11 | \( 1 + 49.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-1.43 + 0.828i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (20.6 + 35.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-130. - 75.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 131. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (136. + 78.8i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (15.9 + 9.20i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-173. + 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (16.9 + 29.4i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-29.5 + 51.2i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-108. - 187. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-174. + 100. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-149. + 259. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (69.6 - 40.2i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-128. + 223. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.03e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (711. - 410. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (52.6 + 91.2i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-245. + 424. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (754. - 1.30e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.08e3 - 627. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.10211533745619926980931295746, −13.20644849465188932742836898037, −11.88659617860560747105899154995, −10.95886846379004954557095440700, −9.675470850749277793994121462883, −8.136248478481278743503342246759, −7.23503795870107891087973178535, −5.69896317010297737750895572273, −4.01322867216859429179578177719, −0.66630320777110072510318096166,
1.61756701075804912838841560442, 4.74791866929605197784641343412, 5.69428673563473441368324816928, 7.48233093204190214263879323537, 9.220417190434745636439733815228, 9.985985253023806102524820293838, 11.27822598775955056385521909853, 11.92105624123501798420352805824, 13.40844415314890458556105782317, 14.94768557341953339909748247786
