| L(s) = 1 | + (−11.0 − 11.0i)3-s + (−38.1 − 38.1i)5-s + 34.3·7-s + 242. i·9-s + (45.9 − 45.9i)11-s + (248. − 248. i)13-s + 840. i·15-s + 552.·17-s + (−606. − 606. i)19-s + (−378. − 378. i)21-s + 562.·23-s − 1.27e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−3.37e3 + 3.37e3i)29-s − 1.84e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.304 − 0.304i)5-s + 0.100·7-s + 0.333i·9-s + (0.0344 − 0.0344i)11-s + (0.112 − 0.112i)13-s + 0.248i·15-s + 0.112·17-s + (−0.0883 − 0.0883i)19-s + (−0.0408 − 0.0408i)21-s + 0.0461·23-s − 0.814i·25-s + (0.136 − 0.136i)27-s + (−0.138 + 0.138i)29-s − 0.620i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1303365242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1303365242\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (38.1 + 38.1i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 34.3T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-45.9 + 45.9i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (-248. + 248. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 552.T + 2.41e7T^{2} \) |
| 19 | \( 1 + (606. + 606. i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 562.T + 1.48e8T^{2} \) |
| 29 | \( 1 + (3.37e3 - 3.37e3i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 1.84e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-2.46e4 - 2.46e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 7.36e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (2.12e4 - 2.12e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.48e5 - 1.48e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.88e5 - 1.88e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-2.17e5 + 2.17e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (3.15e5 + 3.15e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.59e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 6.16e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.07e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (6.13e5 + 6.13e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 3.35e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 9.27e5T + 8.32e11T^{2} \) |
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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
−9.864327034855204112700222448804, −8.715028110593116329793540533985, −7.920639603350471225143011702931, −6.94188166302868568006292774828, −5.95335174783932090979031973129, −4.94405883406784161303926820897, −3.85980446442933129299771960024, −2.42071980497430964978373679368, −1.11492904961567573610768506289, −0.03448615022678890681151972740,
1.40935561493983050387431384901, 2.96182182500815224852993224885, 4.02228932890137637066107697779, 5.05755897492814189245700153987, 6.10408601533393401629374016621, 7.09268361915625876337967590401, 8.086590517123456080896578156986, 9.170561925478663229751194077186, 10.02394918627313345285272817232, 10.98393706211258264498608991591
