| L(s) = 1 | + (0.939 − 0.342i)2-s + (−1.33 − 1.10i)3-s + (0.766 − 0.642i)4-s + (−0.619 − 3.51i)5-s + (−1.63 − 0.578i)6-s + (0.766 + 0.642i)7-s + (0.500 − 0.866i)8-s + (0.572 + 2.94i)9-s + (−1.78 − 3.08i)10-s + (0.0584 − 0.331i)11-s + (−1.73 + 0.0150i)12-s + (−4.08 − 1.48i)13-s + (0.939 + 0.342i)14-s + (−3.04 + 5.37i)15-s + (0.173 − 0.984i)16-s + (−1.39 − 2.40i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.771 − 0.636i)3-s + (0.383 − 0.321i)4-s + (−0.276 − 1.57i)5-s + (−0.666 − 0.236i)6-s + (0.289 + 0.242i)7-s + (0.176 − 0.306i)8-s + (0.190 + 0.981i)9-s + (−0.563 − 0.976i)10-s + (0.0176 − 0.0999i)11-s + (−0.499 + 0.00433i)12-s + (−1.13 − 0.412i)13-s + (0.251 + 0.0914i)14-s + (−0.785 + 1.38i)15-s + (0.0434 − 0.246i)16-s + (−0.337 − 0.584i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.428652 - 1.23932i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428652 - 1.23932i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.33 + 1.10i)T \) |
| 7 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (0.619 + 3.51i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.0584 + 0.331i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (4.08 + 1.48i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.39 + 2.40i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.27 - 3.93i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.51 + 5.46i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (5.34 - 1.94i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.31 + 3.61i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.41 - 4.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.32 - 1.93i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.04 + 11.5i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.48 - 6.28i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 + (1.93 + 10.9i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.50 - 4.61i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.23 + 0.815i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.40 - 4.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.04 - 1.81i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.47 - 1.99i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.01 + 0.368i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.81 + 13.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.519 - 2.94i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.37656495712661694504792888702, −10.34570010705282044116412686505, −9.113413575150535087250529004615, −8.098223779551494046504739288039, −7.15253281107959637984901713217, −5.84236772502509577244656679204, −5.04229851940068679094647602182, −4.39327862870826701545229642954, −2.32038444860078469599497599476, −0.78756119650340167482463725086,
2.63762423406921749760007248106, 3.85192730335263236108460355393, 4.79131712983993436514468199029, 5.96994847279077582358249482492, 6.95400762797216097781562356743, 7.42554248101472889235512285823, 9.164782916625382334383012729594, 10.23363877835814643714953133034, 11.08404577064305588555018071541, 11.40037282703796263605646123235
