| L(s) = 1 | + (−0.936 + 0.349i)2-s + (2.58 − 1.41i)3-s + (0.755 − 0.654i)4-s + (−2.20 − 0.348i)5-s + (−1.93 + 2.22i)6-s + (−2.57 − 3.44i)7-s + (−0.479 + 0.877i)8-s + (3.08 − 4.79i)9-s + (2.19 − 0.445i)10-s + (2.62 − 1.19i)11-s + (1.03 − 2.76i)12-s + (−0.876 − 0.656i)13-s + (3.61 + 2.32i)14-s + (−6.21 + 2.21i)15-s + (0.142 − 0.989i)16-s + (0.491 + 6.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.662 + 0.247i)2-s + (1.49 − 0.816i)3-s + (0.377 − 0.327i)4-s + (−0.987 − 0.155i)5-s + (−0.788 + 0.909i)6-s + (−0.974 − 1.30i)7-s + (−0.169 + 0.310i)8-s + (1.02 − 1.59i)9-s + (0.692 − 0.140i)10-s + (0.789 − 0.360i)11-s + (0.297 − 0.797i)12-s + (−0.243 − 0.182i)13-s + (0.967 + 0.621i)14-s + (−1.60 + 0.572i)15-s + (0.0355 − 0.247i)16-s + (0.119 + 1.66i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.908975 - 0.713742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.908975 - 0.713742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.936 - 0.349i)T \) |
| 5 | \( 1 + (2.20 + 0.348i)T \) |
| 23 | \( 1 + (-2.87 + 3.83i)T \) |
| good | 3 | \( 1 + (-2.58 + 1.41i)T + (1.62 - 2.52i)T^{2} \) |
| 7 | \( 1 + (2.57 + 3.44i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-2.62 + 1.19i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.876 + 0.656i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-0.491 - 6.87i)T + (-16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (-0.238 - 0.274i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-7.29 - 6.31i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 0.682i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.182 + 0.0397i)T + (33.6 - 15.3i)T^{2} \) |
| 41 | \( 1 + (1.27 - 0.818i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (4.01 + 7.35i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (2.00 + 2.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.64 - 2.72i)T + (14.9 - 50.8i)T^{2} \) |
| 59 | \( 1 + (-7.97 + 1.14i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.0664 - 0.226i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.631 + 1.69i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (2.82 - 6.17i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.68 - 0.120i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-2.16 - 15.0i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.16 - 9.93i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (16.0 + 4.72i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.00 + 9.23i)T + (-88.2 - 40.2i)T^{2} \) |
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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
−12.30534672273100477532554797653, −10.78775906478150757512840339559, −9.866232670789361920427390402500, −8.626869154751826724240019135864, −8.240267158764245672738561865917, −7.06563988096549263910940472506, −6.66134481949416031306376190586, −4.01533122458925708890477706358, −3.14638308845917698401324621367, −1.10294832882850962319942736682,
2.64685127266367366447348701288, 3.30223258643151457970651029637, 4.66268175802487905813157019977, 6.72125417677893403569494698926, 7.81670956067629970672045154637, 8.772658866461293844664237232244, 9.408717848869886460649263535514, 9.957257197308362151380837328656, 11.53806170425138239658335361757, 12.11821930997716385294586316599
