| L(s) = 1 | + (−1.28 + 0.587i)2-s + (0.576 + 1.96i)3-s + (1.30 − 1.51i)4-s + (1.83 − 4.65i)5-s + (−1.89 − 2.18i)6-s + (−0.457 − 3.18i)7-s + (−0.796 + 2.71i)8-s + (4.04 − 2.59i)9-s + (0.370 + 7.06i)10-s + (−18.0 − 8.26i)11-s + (3.72 + 1.70i)12-s + (−14.1 − 2.03i)13-s + (2.45 + 3.82i)14-s + (10.1 + 0.924i)15-s + (−0.569 − 3.95i)16-s + (−8.56 − 9.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 + 0.293i)2-s + (0.192 + 0.654i)3-s + (0.327 − 0.377i)4-s + (0.367 − 0.930i)5-s + (−0.316 − 0.364i)6-s + (−0.0653 − 0.454i)7-s + (−0.0996 + 0.339i)8-s + (0.449 − 0.288i)9-s + (0.0370 + 0.706i)10-s + (−1.64 − 0.751i)11-s + (0.310 + 0.141i)12-s + (−1.08 − 0.156i)13-s + (0.175 + 0.273i)14-s + (0.679 + 0.0616i)15-s + (−0.0355 − 0.247i)16-s + (−0.503 − 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.699006 - 0.564534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.699006 - 0.564534i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 - 0.587i)T \) |
| 5 | \( 1 + (-1.83 + 4.65i)T \) |
| 23 | \( 1 + (-22.0 + 6.55i)T \) |
| good | 3 | \( 1 + (-0.576 - 1.96i)T + (-7.57 + 4.86i)T^{2} \) |
| 7 | \( 1 + (0.457 + 3.18i)T + (-47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (18.0 + 8.26i)T + (79.2 + 91.4i)T^{2} \) |
| 13 | \( 1 + (14.1 + 2.03i)T + (162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (8.56 + 9.88i)T + (-41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (4.81 + 4.16i)T + (51.3 + 357. i)T^{2} \) |
| 29 | \( 1 + (15.7 + 18.1i)T + (-119. + 832. i)T^{2} \) |
| 31 | \( 1 + (-50.2 - 14.7i)T + (808. + 519. i)T^{2} \) |
| 37 | \( 1 + (-23.8 + 15.2i)T + (568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-30.5 - 19.6i)T + (698. + 1.52e3i)T^{2} \) |
| 43 | \( 1 + (77.6 - 22.7i)T + (1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 + 61.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (10.8 + 75.6i)T + (-2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (7.33 - 51.0i)T + (-3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (14.8 - 50.6i)T + (-3.13e3 - 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-28.2 - 61.8i)T + (-2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-43.9 - 96.1i)T + (-3.30e3 + 3.80e3i)T^{2} \) |
| 73 | \( 1 + (16.7 + 14.5i)T + (758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-15.8 - 2.27i)T + (5.98e3 + 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-31.8 + 20.4i)T + (2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (41.3 + 140. i)T + (-6.66e3 + 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-52.8 - 33.9i)T + (3.90e3 + 8.55e3i)T^{2} \) |
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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
−11.58336596329063608727737686998, −10.33239759289886387302828635382, −9.899898847924387088782637626875, −8.876474443557805644486811123927, −8.009420841702358248107262467669, −6.86856677510412000432739737829, −5.35993570198145616782440219627, −4.56695035678381900561391220227, −2.67718194723273639133435127724, −0.55493098294209301503889307424,
2.03980572596150157846753004603, 2.76580002790044702020835840981, 4.87870127335348808168604871548, 6.42966366806670136543695722079, 7.40091341859586447311298535260, 7.990930819127984840513408885844, 9.468530734648578340451131347882, 10.26763761290691454361071301665, 10.97097980935126175347452512755, 12.31154588885547498260753012049
