| L(s) = 1 | + (0.939 − 0.342i)2-s + (1.32 + 1.12i)3-s + (0.766 − 0.642i)4-s + (0.359 + 2.03i)5-s + (1.62 + 0.601i)6-s + (0.766 + 0.642i)7-s + (0.500 − 0.866i)8-s + (0.488 + 2.95i)9-s + (1.03 + 1.79i)10-s + (0.153 − 0.869i)11-s + (1.73 + 0.00936i)12-s + (−6.62 − 2.41i)13-s + (0.939 + 0.342i)14-s + (−1.80 + 3.09i)15-s + (0.173 − 0.984i)16-s + (−0.458 − 0.793i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.762 + 0.646i)3-s + (0.383 − 0.321i)4-s + (0.160 + 0.910i)5-s + (0.663 + 0.245i)6-s + (0.289 + 0.242i)7-s + (0.176 − 0.306i)8-s + (0.162 + 0.986i)9-s + (0.327 + 0.566i)10-s + (0.0462 − 0.262i)11-s + (0.499 + 0.00270i)12-s + (−1.83 − 0.668i)13-s + (0.251 + 0.0914i)14-s + (−0.466 + 0.798i)15-s + (0.0434 − 0.246i)16-s + (−0.111 − 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.36012 + 0.760875i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.36012 + 0.760875i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.32 - 1.12i)T \) |
| 7 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (-0.359 - 2.03i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.153 + 0.869i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (6.62 + 2.41i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.458 + 0.793i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0604 + 0.104i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.30 + 3.61i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.98 + 2.54i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.33 + 1.12i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.42 + 5.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.97 + 1.08i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.63 - 9.25i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.61 + 3.87i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + (0.366 + 2.07i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.69 - 3.09i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (13.2 + 4.80i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.0193 - 0.0335i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.28 - 10.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.01 - 2.55i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.691 + 0.251i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.567 + 0.982i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.809 + 4.59i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.38074203019428321655860183350, −10.40413216090977410116137017368, −9.971939347794437471672946615278, −8.767239902192943650033010945336, −7.64381547417206530022561742930, −6.73025718905608666854700587930, −5.29795008498506804834892086857, −4.49112186627701995586523085382, −3.00624409083624127975222329775, −2.47873520872896885075077076846,
1.60356993726398140621843709497, 2.94355293985829898534688896915, 4.45173154754198167877272776271, 5.18925639457132259792879269321, 6.75518401898008482040776827098, 7.34488530980316809817587169420, 8.425586597052011400809815080443, 9.213864640813748776246728255213, 10.23532740741886177902024411189, 11.88161841483857550584232392835
