| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.0130 − 0.0909i)3-s + (0.841 − 0.540i)4-s + (−1.56 + 3.43i)5-s + (0.0381 + 0.0835i)6-s + (−2.03 + 0.598i)7-s + (−0.654 + 0.755i)8-s + (2.87 − 0.842i)9-s + (0.537 − 3.73i)10-s + (−1.78 + 3.91i)11-s + (−0.0601 − 0.0694i)12-s + (2.58 + 2.97i)13-s + (1.78 − 1.14i)14-s + (0.332 + 0.0976i)15-s + (0.415 − 0.909i)16-s + (−1.95 − 1.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.00754 − 0.0524i)3-s + (0.420 − 0.270i)4-s + (−0.701 + 1.53i)5-s + (0.0155 + 0.0341i)6-s + (−0.770 + 0.226i)7-s + (−0.231 + 0.267i)8-s + (0.956 − 0.280i)9-s + (0.169 − 1.18i)10-s + (−0.539 + 1.18i)11-s + (−0.0173 − 0.0200i)12-s + (0.715 + 0.825i)13-s + (0.477 − 0.307i)14-s + (0.0858 + 0.0252i)15-s + (0.103 − 0.227i)16-s + (−0.474 − 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.415572 + 0.497211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415572 + 0.497211i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 67 | \( 1 + (-7.58 + 3.08i)T \) |
| good | 3 | \( 1 + (0.0130 + 0.0909i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (1.56 - 3.43i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (2.03 - 0.598i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.78 - 3.91i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-2.58 - 2.97i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.95 + 1.25i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.770 + 0.226i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (0.0694 + 0.482i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 - 9.13T + 29T^{2} \) |
| 31 | \( 1 + (-5.95 + 6.87i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + (0.800 + 0.514i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (8.87 + 5.70i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.08 - 7.51i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (0.509 - 0.327i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.04 + 4.66i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-6.33 - 13.8i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (-2.82 + 1.81i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.21 - 9.22i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.20 - 4.84i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (2.72 - 5.96i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.178 + 1.23i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−13.59314108895387078892747690429, −12.31323910479529821565633537315, −11.33462630954051682349931441406, −10.26387005278490356914700109000, −9.636034118778015699071696175595, −8.071643532785431127273112482089, −6.85981459247094592891872879743, −6.58772167752987356123765907349, −4.18719725502436256480993123729, −2.57470362623026273666995067960,
0.875032391957450775486677655894, 3.48166591177200187553757429109, 4.91784390924715909547194373645, 6.52598807643018433322243917264, 8.170412770528905219110735805981, 8.480767846481165306661113125579, 9.860139188067265429865142315277, 10.76113977270449457365729377979, 12.00330012780299968742810414247, 12.96560040443610778922153042799
