| L(s) = 1 | + (−2.75 − 0.660i)2-s + (2.16 − 4.72i)3-s + (7.12 + 3.63i)4-s + (−3.60 + 9.91i)5-s + (−9.06 + 11.5i)6-s + (−12.1 − 2.14i)7-s + (−17.2 − 14.6i)8-s + (−17.6 − 20.4i)9-s + (16.4 − 24.8i)10-s + (−29.6 + 10.7i)11-s + (32.5 − 25.8i)12-s + (7.47 − 6.26i)13-s + (32.0 + 13.9i)14-s + (39.0 + 38.4i)15-s + (37.6 + 51.7i)16-s + (−107. + 61.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 − 0.233i)2-s + (0.416 − 0.909i)3-s + (0.890 + 0.454i)4-s + (−0.322 + 0.887i)5-s + (−0.616 + 0.787i)6-s + (−0.658 − 0.116i)7-s + (−0.760 − 0.649i)8-s + (−0.653 − 0.756i)9-s + (0.521 − 0.787i)10-s + (−0.812 + 0.295i)11-s + (0.783 − 0.621i)12-s + (0.159 − 0.133i)13-s + (0.612 + 0.266i)14-s + (0.672 + 0.662i)15-s + (0.587 + 0.809i)16-s + (−1.52 + 0.881i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0309788 + 0.0764939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0309788 + 0.0764939i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.75 + 0.660i)T \) |
| 3 | \( 1 + (-2.16 + 4.72i)T \) |
| good | 5 | \( 1 + (3.60 - 9.91i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (12.1 + 2.14i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (29.6 - 10.7i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-7.47 + 6.26i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (107. - 61.8i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (38.1 + 22.0i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-0.485 - 2.75i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (115. - 137. i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-44.5 + 7.85i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (43.4 + 75.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-263. - 314. i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-8.41 - 23.1i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-108. + 616. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 438. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (605. + 220. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (14.7 - 83.6i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (61.6 + 73.4i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (383. + 663. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-336. + 582. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (601. - 716. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-1.02e3 - 858. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (184. + 106. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.23e3 - 450. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.25765281816540433131014243217, −12.64540901449052364779022124418, −11.27489297070082099498117214651, −10.53461246326985373407129750616, −9.182471831759066262777199013781, −8.125988136014540877839132676329, −7.07677710290985666021224383139, −6.37080259712941344385588860357, −3.38420117624877165198588055904, −2.17551928979621279742141284026,
0.05364697632848580791423852859, 2.64136906631987389929813745938, 4.53785631397296370627837045590, 5.97718913245371305160723479634, 7.65079078948061266151062342023, 8.764634309029360907217660885703, 9.305950960020321912883009882158, 10.48753770030990755969496198908, 11.39567103189197659068892676340, 12.79997637286812673384285454757
