| L(s) = 1 | + (1.43 − 1.20i)3-s + (3.31 + 1.20i)5-s + (−0.826 − 0.300i)7-s + (0.0923 − 0.524i)9-s + (1.67 − 2.89i)11-s + (−1.11 − 6.31i)13-s + (6.23 − 2.27i)15-s + (−0.520 + 2.95i)17-s + (−3.55 + 2.98i)19-s + (−1.55 + 0.565i)21-s + (2.91 + 5.05i)23-s + (5.72 + 4.80i)25-s + (2.31 + 4.01i)27-s + (1.63 − 2.83i)29-s + 1.12·31-s + ⋯ |
| L(s) = 1 | + (0.831 − 0.697i)3-s + (1.48 + 0.540i)5-s + (−0.312 − 0.113i)7-s + (0.0307 − 0.174i)9-s + (0.504 − 0.874i)11-s + (−0.308 − 1.75i)13-s + (1.61 − 0.586i)15-s + (−0.126 + 0.716i)17-s + (−0.815 + 0.683i)19-s + (−0.338 + 0.123i)21-s + (0.608 + 1.05i)23-s + (1.14 + 0.961i)25-s + (0.446 + 0.773i)27-s + (0.303 − 0.525i)29-s + 0.201·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25256 - 0.640320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25256 - 0.640320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (3.44 + 5.01i)T \) |
| good | 3 | \( 1 + (-1.43 + 1.20i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.31 - 1.20i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.826 + 0.300i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.67 + 2.89i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.11 + 6.31i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.520 - 2.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (3.55 - 2.98i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.91 - 5.05i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.63 + 2.83i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 41 | \( 1 + (-1.49 - 8.45i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 + (2.56 + 4.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.252 - 0.0918i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-7.15 + 2.60i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.369 - 2.09i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.01 - 2.91i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (10.0 - 8.45i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 8.57T + 73T^{2} \) |
| 79 | \( 1 + (10.8 + 3.96i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.724 - 4.10i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (7.86 - 2.86i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.53 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.33928724381530467520422377941, −9.880165550249941224211837507802, −8.701519439512387562230055724386, −8.084692881992062071729297586529, −6.98787862900620066535563289723, −6.09871672893451493426131446726, −5.41064324065573300423027478912, −3.45387271138091834521201437581, −2.64075309007320788228538459151, −1.50589785911462737464270583775,
1.81258045178044227109318279009, 2.76479207262823042253124285365, 4.34081596553181111784851599697, 4.90089089907987623496107247415, 6.41021137753055201719523600966, 6.91200227113979262004418604875, 8.789932412134245849330310571994, 9.009028614823915811885498186872, 9.695260137480847561914011806574, 10.32097857545916180850588995262
