| L(s) = 1 | + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.939 − 0.342i)6-s + (1.26 − 2.19i)7-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.766 − 0.642i)10-s + (0.326 + 0.565i)11-s + (0.499 − 0.866i)12-s + (0.213 − 0.0775i)13-s + (−0.439 − 2.49i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.624 − 0.524i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (0.383 − 0.139i)6-s + (0.478 − 0.828i)7-s + (−0.176 − 0.306i)8-s + (0.255 + 0.214i)9-s + (−0.242 − 0.203i)10-s + (0.0983 + 0.170i)11-s + (0.144 − 0.250i)12-s + (0.0590 − 0.0215i)13-s + (−0.117 − 0.666i)14-s + (0.0448 − 0.254i)15-s + (−0.234 − 0.0855i)16-s + (0.151 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92664 - 1.38162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92664 - 1.38162i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-0.0320 + 4.35i)T \) |
| good | 7 | \( 1 + (-1.26 + 2.19i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.326 - 0.565i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.213 + 0.0775i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.624 + 0.524i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.680 - 3.86i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.14 + 0.957i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.479 + 0.829i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.34T + 37T^{2} \) |
| 41 | \( 1 + (-4.35 - 1.58i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.273 + 1.55i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.06 - 5.09i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (1.06 - 6.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (2.37 - 1.99i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.553 - 3.13i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.690 + 0.579i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 9.09i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (10.1 + 3.70i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.69 - 0.979i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (4.46 - 7.72i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (14.2 - 5.16i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.603 - 0.506i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.69345229701103839931260053738, −9.708949612369753894662471548331, −8.991406303327854569601370347328, −7.85193410571056267442191432251, −7.09821252408074153951850973960, −5.72030487054198995948211613008, −4.61785448965249594221974898318, −3.97633248416355936004184759879, −2.69378939598072108606621133740, −1.23317445215609782005106968270,
2.02371678050525127963166875003, 3.16621267200114334095285781343, 4.25601874714169687916954228697, 5.49851208904861705538393696333, 6.31599356081470353301855938591, 7.35265078187624474421604418783, 8.209806197523841214002002481271, 8.850565132239026850525530212588, 9.997152352366252891308546169038, 11.03733047123603756088816764786
