| L(s) = 1 | − 0.219·2-s − 1.60·3-s − 1.95·4-s − 5-s + 0.351·6-s − 0.332·7-s + 0.868·8-s − 0.439·9-s + 0.219·10-s − 5.37·11-s + 3.12·12-s + 0.0729·14-s + 1.60·15-s + 3.71·16-s − 5.06·17-s + 0.0965·18-s + 2.26·19-s + 1.95·20-s + 0.531·21-s + 1.18·22-s − 2.83·23-s − 1.38·24-s + 25-s + 5.50·27-s + 0.648·28-s − 2.90·29-s − 0.351·30-s + ⋯ |
| L(s) = 1 | − 0.155·2-s − 0.923·3-s − 0.975·4-s − 0.447·5-s + 0.143·6-s − 0.125·7-s + 0.306·8-s − 0.146·9-s + 0.0694·10-s − 1.61·11-s + 0.901·12-s + 0.0195·14-s + 0.413·15-s + 0.928·16-s − 1.22·17-s + 0.0227·18-s + 0.520·19-s + 0.436·20-s + 0.116·21-s + 0.251·22-s − 0.592·23-s − 0.283·24-s + 0.200·25-s + 1.05·27-s + 0.122·28-s − 0.539·29-s − 0.0641·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3694386615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3694386615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.219T + 2T^{2} \) |
| 3 | \( 1 + 1.60T + 3T^{2} \) |
| 7 | \( 1 + 0.332T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 5.06T + 43T^{2} \) |
| 47 | \( 1 - 8.34T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 + 4.26T + 83T^{2} \) |
| 89 | \( 1 - 3.22T + 89T^{2} \) |
| 97 | \( 1 + 2.50T + 97T^{2} \) |
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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
−10.26776355761131396925915645709, −9.438996537018788798496388404267, −8.369500632942397135431649782756, −7.86570259510938438699083825317, −6.66726904731109414608382943365, −5.57811051604938807503576349496, −4.98867789305390063232823533653, −4.05262442453362287767774656689, −2.65682111490733906699734462289, −0.50550891593261280958217540002,
0.50550891593261280958217540002, 2.65682111490733906699734462289, 4.05262442453362287767774656689, 4.98867789305390063232823533653, 5.57811051604938807503576349496, 6.66726904731109414608382943365, 7.86570259510938438699083825317, 8.369500632942397135431649782756, 9.438996537018788798496388404267, 10.26776355761131396925915645709
