| L(s) = 1 | + (−0.698 + 1.22i)2-s + i·3-s + (−1.02 − 1.71i)4-s + i·5-s + (−1.22 − 0.698i)6-s + 0.344·7-s + (2.82 − 0.0624i)8-s − 9-s + (−1.22 − 0.698i)10-s + 1.11·11-s + (1.71 − 1.02i)12-s − 4.23·13-s + (−0.240 + 0.423i)14-s − 15-s + (−1.89 + 3.52i)16-s − 3.31i·17-s + ⋯ |
| L(s) = 1 | + (−0.493 + 0.869i)2-s + 0.577i·3-s + (−0.512 − 0.858i)4-s + 0.447i·5-s + (−0.502 − 0.284i)6-s + 0.130·7-s + (0.999 − 0.0220i)8-s − 0.333·9-s + (−0.388 − 0.220i)10-s + 0.336·11-s + (0.495 − 0.296i)12-s − 1.17·13-s + (−0.0642 + 0.113i)14-s − 0.258·15-s + (−0.474 + 0.880i)16-s − 0.804i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1234796921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1234796921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.698 - 1.22i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (4.54 + 1.51i)T \) |
| good | 7 | \( 1 - 0.344T + 7T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 29 | \( 1 - 0.696T + 29T^{2} \) |
| 31 | \( 1 + 1.78iT - 31T^{2} \) |
| 37 | \( 1 - 8.80iT - 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + 8.35iT - 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 + 14.6iT - 59T^{2} \) |
| 61 | \( 1 + 7.06iT - 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 + 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 2.68T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 11.9iT - 89T^{2} \) |
| 97 | \( 1 - 9.50iT - 97T^{2} \) |
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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
−9.626473113844685737563903331282, −8.480615457425713296594910572893, −7.88183318952495517026526869538, −6.88721519373423319399480790628, −6.33744436984803381645936410736, −5.13638803558641910460827969211, −4.65348665940252145760774772948, −3.38995407835317323989195550616, −1.99960923228611675166218480500, −0.05846939894584919814804420859,
1.46598177155223127932607818367, 2.30989636508254294698529451289, 3.54912269543559808961035109690, 4.51100955451424519398698174445, 5.48327028594716793477594239557, 6.70093694259742583678588733901, 7.56080724036908377573559779476, 8.286554959756218151156069755370, 8.917289725677189429802732897684, 9.867342215735136239470557995490
