| L(s) = 1 | + (0.994 − 0.104i)2-s + (−0.309 + 0.951i)3-s + (0.978 − 0.207i)4-s + (−0.207 − 0.978i)5-s + (−0.207 + 0.978i)6-s + (−0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.169 − 1.60i)11-s + (−0.104 + 0.994i)12-s + (0.614 + 0.0646i)13-s + (−0.913 − 0.406i)14-s + (0.994 + 0.104i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + ⋯ |
| L(s) = 1 | + (0.994 − 0.104i)2-s + (−0.309 + 0.951i)3-s + (0.978 − 0.207i)4-s + (−0.207 − 0.978i)5-s + (−0.207 + 0.978i)6-s + (−0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.169 − 1.60i)11-s + (−0.104 + 0.994i)12-s + (0.614 + 0.0646i)13-s + (−0.913 − 0.406i)14-s + (0.994 + 0.104i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.703451709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.703451709\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.994 + 0.104i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.207 + 0.978i)T \) |
| good | 7 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.169 + 1.60i)T + (-0.978 + 0.207i)T^{2} \) |
| 13 | \( 1 + (-0.614 - 0.0646i)T + (0.978 + 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \) |
| 29 | \( 1 + (-1.20 + 1.08i)T + (0.104 - 0.994i)T^{2} \) |
| 31 | \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1.20 - 1.08i)T + (0.104 - 0.994i)T^{2} \) |
| 53 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 61 | \( 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2} \) |
| 67 | \( 1 + (0.413 - 0.459i)T + (-0.104 - 0.994i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (1.20 - 1.08i)T + (0.104 - 0.994i)T^{2} \) |
| 83 | \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
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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
−9.635482761818504859933627429481, −8.464668750171277603542392936135, −8.042412895286337764135048370858, −6.42956266366312262041439704789, −6.03591374253767507047815609064, −5.29963388165344574564409447386, −4.25488652292383696083513033767, −3.72571272041449654510938818796, −2.98162193555925566154476841822, −1.01704448559718715821791710628,
1.83529047845072113410057831389, 2.81575054736681495161565192729, 3.40817303349792559497735728041, 4.87910854477053017773897100562, 5.52686823648032189695844344521, 6.55845966979045849029671967730, 7.08494872540884584937711936678, 7.32467245495797044802074275709, 8.565425242932211818579640400127, 9.708102488544302382016441377958
