| L(s) = 1 | + (0.607 − 2.66i)2-s + (6.07 − 2.92i)3-s + (0.498 + 0.239i)4-s + (−1.74 + 11.0i)5-s + (−4.09 − 17.9i)6-s + (12.3 + 25.5i)7-s + (14.5 − 18.2i)8-s + (11.5 − 14.4i)9-s + (28.3 + 11.3i)10-s + (6.40 − 5.10i)11-s + 3.72·12-s + (39.6 − 31.6i)13-s + (75.5 − 17.2i)14-s + (21.6 + 72.2i)15-s + (−36.9 − 46.3i)16-s − 76.2·17-s + ⋯ |
| L(s) = 1 | + (0.214 − 0.940i)2-s + (1.16 − 0.563i)3-s + (0.0622 + 0.0299i)4-s + (−0.156 + 0.987i)5-s + (−0.278 − 1.22i)6-s + (0.665 + 1.38i)7-s + (0.643 − 0.806i)8-s + (0.427 − 0.536i)9-s + (0.895 + 0.359i)10-s + (0.175 − 0.140i)11-s + 0.0897·12-s + (0.846 − 0.675i)13-s + (1.44 − 0.329i)14-s + (0.373 + 1.24i)15-s + (−0.577 − 0.724i)16-s − 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.80142 - 1.25293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.80142 - 1.25293i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.74 - 11.0i)T \) |
| 29 | \( 1 + (133. + 81.7i)T \) |
| good | 2 | \( 1 + (-0.607 + 2.66i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-6.07 + 2.92i)T + (16.8 - 21.1i)T^{2} \) |
| 7 | \( 1 + (-12.3 - 25.5i)T + (-213. + 268. i)T^{2} \) |
| 11 | \( 1 + (-6.40 + 5.10i)T + (296. - 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-39.6 + 31.6i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + 76.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-14.4 + 29.9i)T + (-4.27e3 - 5.36e3i)T^{2} \) |
| 23 | \( 1 + (132. - 30.3i)T + (1.09e4 - 5.27e3i)T^{2} \) |
| 31 | \( 1 + (-43.7 - 9.99i)T + (2.68e4 + 1.29e4i)T^{2} \) |
| 37 | \( 1 + (-176. + 221. i)T + (-1.12e4 - 4.93e4i)T^{2} \) |
| 41 | \( 1 + 180. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (61.7 + 270. i)T + (-7.16e4 + 3.44e4i)T^{2} \) |
| 47 | \( 1 + (-243. - 305. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-211. - 48.2i)T + (1.34e5 + 6.45e4i)T^{2} \) |
| 59 | \( 1 + 901.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (170. + 354. i)T + (-1.41e5 + 1.77e5i)T^{2} \) |
| 67 | \( 1 + (-198. - 158. i)T + (6.69e4 + 2.93e5i)T^{2} \) |
| 71 | \( 1 + (-297. - 372. i)T + (-7.96e4 + 3.48e5i)T^{2} \) |
| 73 | \( 1 + (-266. - 1.16e3i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + (497. + 396. i)T + (1.09e5 + 4.80e5i)T^{2} \) |
| 83 | \( 1 + (-222. + 462. i)T + (-3.56e5 - 4.47e5i)T^{2} \) |
| 89 | \( 1 + (-921. - 210. i)T + (6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + (-405. - 195. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−12.43196308815413983443887924206, −11.45402110095087941895838143427, −10.80878431049402496398902840769, −9.332231126403513775357210923219, −8.285941539812005158984544368976, −7.38452864157050548031391419937, −5.98585626029669528199248688277, −3.81886432929636701195581540744, −2.64551804170092131936827729670, −1.99841561201348460952647136253,
1.66704158387018857092819079601, 3.99555734473400118070314609022, 4.64052174901247051581761392892, 6.34081925289054216804076658389, 7.69573719677788383916310701962, 8.315292900267019987987992499419, 9.333436692635925416961737722882, 10.61098251480081147221214236440, 11.65746834284655026828303423748, 13.47265864025485915010135698696
