| L(s) = 1 | + (−0.398 + 0.827i)2-s + (−2.34 − 1.87i)3-s + (4.46 + 5.59i)4-s + (−12.2 − 5.87i)5-s + (2.48 − 1.19i)6-s + (20.2 − 25.3i)7-s + (−13.5 + 3.09i)8-s + (2.00 + 8.77i)9-s + (9.72 − 7.75i)10-s + (25.6 + 5.86i)11-s − 21.4i·12-s + (17.6 − 77.4i)13-s + (12.9 + 26.8i)14-s + (17.6 + 36.6i)15-s + (−9.89 + 43.3i)16-s − 81.2i·17-s + ⋯ |
| L(s) = 1 | + (−0.140 + 0.292i)2-s + (−0.451 − 0.359i)3-s + (0.557 + 0.699i)4-s + (−1.09 − 0.525i)5-s + (0.168 − 0.0813i)6-s + (1.09 − 1.36i)7-s + (−0.599 + 0.136i)8-s + (0.0741 + 0.324i)9-s + (0.307 − 0.245i)10-s + (0.704 + 0.160i)11-s − 0.516i·12-s + (0.377 − 1.65i)13-s + (0.246 + 0.511i)14-s + (0.303 + 0.630i)15-s + (−0.154 + 0.677i)16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.05721 - 0.552297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05721 - 0.552297i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.34 + 1.87i)T \) |
| 29 | \( 1 + (43.3 - 150. i)T \) |
| good | 2 | \( 1 + (0.398 - 0.827i)T + (-4.98 - 6.25i)T^{2} \) |
| 5 | \( 1 + (12.2 + 5.87i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (-20.2 + 25.3i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (-25.6 - 5.86i)T + (1.19e3 + 577. i)T^{2} \) |
| 13 | \( 1 + (-17.6 + 77.4i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + 81.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-30.1 + 24.0i)T + (1.52e3 - 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-28.4 + 13.6i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (56.5 - 117. i)T + (-1.85e4 - 2.32e4i)T^{2} \) |
| 37 | \( 1 + (164. - 37.4i)T + (4.56e4 - 2.19e4i)T^{2} \) |
| 41 | \( 1 + 137. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-86.7 - 180. i)T + (-4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (295. + 67.5i)T + (9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-457. - 220. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 - 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-384. - 306. i)T + (5.05e4 + 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-24.7 - 108. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-187. + 822. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-170. - 354. i)T + (-2.42e5 + 3.04e5i)T^{2} \) |
| 79 | \( 1 + (-1.04e3 + 238. i)T + (4.44e5 - 2.13e5i)T^{2} \) |
| 83 | \( 1 + (610. + 765. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-224. + 466. i)T + (-4.39e5 - 5.51e5i)T^{2} \) |
| 97 | \( 1 + (1.18e3 - 945. i)T + (2.03e5 - 8.89e5i)T^{2} \) |
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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
−13.37739557822041704091151039642, −12.26843493021698153852689911427, −11.47271213398053062426994282710, −10.66913400449083433324640862726, −8.548552836858462222404678377566, −7.63961734566367130772566110372, −7.04911435520940959979691045511, −5.02706202157263492814163778782, −3.60599249004516344676961660029, −0.853962283160166895595571612397,
1.86258008835870574448966745870, 3.97672337057502204450269884560, 5.61025827819470670802076876609, 6.75501788520560178924813624192, 8.382698337179828866324903224823, 9.523704724276083524527035916003, 11.11125793191741214656896409780, 11.48580794846860506125664392651, 12.09579941925004773830164906398, 14.34452764254718643424974303155
