L(s) = 1 | + (−1 − 3.87i)2-s + 6·3-s + (−14.0 + 7.74i)4-s + 30.9i·5-s + (−6 − 23.2i)6-s − 61.9i·7-s + (44.0 + 46.4i)8-s − 45·9-s + (120. − 30.9i)10-s − 26·11-s + (−84.0 + 46.4i)12-s − 30.9i·13-s + (−240. + 61.9i)14-s + 185. i·15-s + (136. − 216. i)16-s + 226·17-s + ⋯ |
L(s) = 1 | + (−0.250 − 0.968i)2-s + 0.666·3-s + (−0.875 + 0.484i)4-s + 1.23i·5-s + (−0.166 − 0.645i)6-s − 1.26i·7-s + (0.687 + 0.726i)8-s − 0.555·9-s + (1.20 − 0.309i)10-s − 0.214·11-s + (−0.583 + 0.322i)12-s − 0.183i·13-s + (−1.22 + 0.316i)14-s + 0.826i·15-s + (0.531 − 0.847i)16-s + 0.782·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.855528 - 0.368160i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.855528 - 0.368160i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 3.87i)T \) |
good | 3 | \( 1 - 6T + 81T^{2} \) |
| 5 | \( 1 - 30.9iT - 625T^{2} \) |
| 7 | \( 1 + 61.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 26T + 1.46e4T^{2} \) |
| 13 | \( 1 + 30.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 226T + 8.35e4T^{2} \) |
| 19 | \( 1 - 134T + 1.30e5T^{2} \) |
| 23 | \( 1 - 309. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 340. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.23e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.76e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 994T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.88e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.10e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.81e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.01e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.07e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 8.00e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 557. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 386T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.10e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 2.23e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.00e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 8.73e3T + 8.85e7T^{2} \) |
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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
−20.75851879792913376932067769058, −19.75554523375872504894024081820, −18.57549211362932464980201114502, −17.11849225917962360839081028424, −14.54604406493841985013454267836, −13.54086326319204164680456829676, −11.25256454262619031776370893028, −9.965584594603764605037194581979, −7.73365253042416185622777892779, −3.27029599977862769934910771922,
5.42938113619192888783297542980, 8.300859782990254159011153936074, 9.223835836731734465687979184813, 12.49505108077315281617686365600, 14.18644411469039162318032379072, 15.59975009337965161486346141718, 16.84746508040984966204166572848, 18.47094474270830168394216425506, 19.88017054250292570081466802057, 21.41974751600567597501507952751