| L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−1.51 + 0.445i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (0.959 + 0.281i)10-s + (2.10 + 2.43i)11-s + (0.654 + 0.755i)12-s + (2.52 + 0.740i)13-s + (−0.656 + 1.43i)14-s + (−0.841 − 0.540i)15-s + (−0.959 + 0.281i)16-s + (0.365 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (−0.0580 + 0.404i)6-s + (−0.573 + 0.168i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (0.303 + 0.0890i)10-s + (0.635 + 0.733i)11-s + (0.189 + 0.218i)12-s + (0.699 + 0.205i)13-s + (−0.175 + 0.384i)14-s + (−0.217 − 0.139i)15-s + (−0.239 + 0.0704i)16-s + (0.0886 − 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57880 + 0.295225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57880 + 0.295225i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.96 - 3.76i)T \) |
| good | 7 | \( 1 + (1.51 - 0.445i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.10 - 2.43i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.52 - 0.740i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 2.54i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.869 - 6.04i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.0775 + 0.539i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.09 - 4.55i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.27 + 4.98i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.64 - 7.97i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.18 + 3.97i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + (6.70 - 1.96i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (5.21 + 1.53i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (11.0 + 7.12i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-9.75 + 11.2i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (2.29 - 2.65i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.103 - 0.718i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-15.2 - 4.47i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.03 - 8.84i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.92 + 5.09i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (0.773 + 1.69i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.65600029099018806723762660689, −9.630196898153297742337289910504, −9.366400745790070024325832282369, −7.83170983197531823859693177802, −6.62056769623417127030013556832, −6.06377392819975793549575266003, −4.97310753871689608140989614309, −3.92319622962291431127720014710, −3.00493889600930361977836947266, −1.45049719199352237981473372818,
0.887373545821947910157716797865, 2.84847914133187067611605844262, 4.07498527377318312214614222066, 5.05757562572841820274361708234, 6.23292416524507538812022907471, 6.45804645228350939982262619924, 7.69017017412652710073640631158, 8.636500732287270608281011201462, 9.359090100828850807654813854917, 10.58496605992849641442649889096
