| L(s) = 1 | + (3.58 − 3.76i)3-s + (−1.13 + 1.96i)5-s + (−14.5 − 11.4i)7-s + (−1.31 − 26.9i)9-s + (−31.0 + 17.9i)11-s + (−70.2 + 40.5i)13-s + (3.33 + 11.3i)15-s + (25.8 − 44.7i)17-s + (−9.61 + 5.55i)19-s + (−95.2 + 13.4i)21-s + (−26.4 − 15.2i)23-s + (59.9 + 103. i)25-s + (−106. − 91.7i)27-s + (−167. − 96.9i)29-s − 188. i·31-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.724i)3-s + (−0.101 + 0.176i)5-s + (−0.783 − 0.620i)7-s + (−0.0486 − 0.998i)9-s + (−0.851 + 0.491i)11-s + (−1.49 + 0.865i)13-s + (0.0574 + 0.195i)15-s + (0.368 − 0.638i)17-s + (−0.116 + 0.0670i)19-s + (−0.990 + 0.139i)21-s + (−0.239 − 0.138i)23-s + (0.479 + 0.830i)25-s + (−0.756 − 0.653i)27-s + (−1.07 − 0.620i)29-s − 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4942348630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4942348630\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.58 + 3.76i)T \) |
| 7 | \( 1 + (14.5 + 11.4i)T \) |
| good | 5 | \( 1 + (1.13 - 1.96i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (31.0 - 17.9i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (70.2 - 40.5i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-25.8 + 44.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (9.61 - 5.55i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (26.4 + 15.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (167. + 96.9i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 188. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-25.3 - 43.9i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-151. - 261. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-28.1 + 48.7i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (548. + 316. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + 686.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 393. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 18.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-458. - 264. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 463.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-212. + 368. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-74.5 - 129. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.05e3 + 611. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.19885444522899016991289388167, −9.739189997216068132845783153484, −9.458056097507047337178102575689, −7.75313952624326479789471125685, −7.35601919017270090484381069592, −6.31601775194822210320863656631, −4.67864122177531730873934995474, −3.25621012556198163755313042985, −2.12532624402145526143682298924, −0.16271379764877894235597367044,
2.48370419645429470629425649459, 3.36870956644619304014895637955, 4.88126706635397882987614026370, 5.77572544291448943035548982567, 7.42087097687797033559731056368, 8.320110273707281364002056119642, 9.254896043349265262367476902425, 10.13180736060340157622341145648, 10.82132300099100212961744399083, 12.40994180286724643877601774944
