| 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
−14.94768557341953339909748247786, −13.40844415314890458556105782317, −11.92105624123501798420352805824, −11.27822598775955056385521909853, −9.985985253023806102524820293838, −9.220417190434745636439733815228, −7.48233093204190214263879323537, −5.69428673563473441368324816928, −4.74791866929605197784641343412, −1.61756701075804912838841560442,
0.66630320777110072510318096166, 4.01322867216859429179578177719, 5.69896317010297737750895572273, 7.23503795870107891087973178535, 8.136248478481278743503342246759, 9.675470850749277793994121462883, 10.95886846379004954557095440700, 11.88659617860560747105899154995, 13.20644849465188932742836898037, 14.10211533745619926980931295746
