| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.858 − 0.990i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (1.10 + 0.708i)6-s + (−1.50 + 3.28i)7-s + (−0.654 + 0.755i)8-s + (0.182 − 1.26i)9-s + (−0.415 − 0.909i)10-s + (3.02 + 0.887i)11-s + (−1.25 − 0.369i)12-s + (1.46 + 3.20i)13-s + (0.514 − 3.57i)14-s + (0.858 − 0.990i)15-s + (0.415 − 0.909i)16-s + (4.17 + 2.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.495 − 0.572i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (0.450 + 0.289i)6-s + (−0.567 + 1.24i)7-s + (−0.231 + 0.267i)8-s + (0.0607 − 0.422i)9-s + (−0.131 − 0.287i)10-s + (0.911 + 0.267i)11-s + (−0.363 − 0.106i)12-s + (0.405 + 0.888i)13-s + (0.137 − 0.956i)14-s + (0.221 − 0.255i)15-s + (0.103 − 0.227i)16-s + (1.01 + 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678572 + 0.323180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678572 + 0.323180i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-4.77 - 0.419i)T \) |
| good | 3 | \( 1 + (0.858 + 0.990i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (1.50 - 3.28i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.02 - 0.887i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.46 - 3.20i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.17 - 2.68i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.977 - 0.627i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.24 - 2.72i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.03 - 4.65i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.136 + 0.948i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.313 + 2.17i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.53 + 6.38i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + (-5.13 + 11.2i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.30 - 7.23i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.74 + 6.63i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (14.8 - 4.37i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (1.03 - 0.304i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (1.97 - 1.26i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.03 - 6.64i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.08 + 14.5i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (1.03 + 1.19i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.0192 - 0.133i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.07449067776667630893766583928, −11.60930356938889308218755264063, −10.31834472111205178138469290114, −9.261419277948384787709177317403, −8.631496664036768128537204645379, −7.01765523770017445844158249500, −6.49512435583172305991747159934, −5.55650610540003983566594938670, −3.42863759465019858082851984491, −1.65027188127619374296363411377,
0.909963232589006324491973554370, 3.33250980892296830789362170597, 4.57201570041613893877553011752, 5.93898701387609755709424209993, 7.17965544208374835188751873583, 8.173293896040608976338167559728, 9.444969823329980807908618019852, 10.14242139552591338570432269317, 10.90563398814960779507952647486, 11.76396655241829626749832467132
