| L(s) = 1 | + 1.61i·3-s − 2.23·5-s + 1.85i·7-s + 0.381·9-s + 5.61·11-s − 2.61i·13-s − 3.61i·15-s + 0.854i·17-s − 0.145·19-s − 3·21-s + i·23-s + 5.00·25-s + 5.47i·27-s + 9.70·29-s + 2.14·31-s + ⋯ |
| L(s) = 1 | + 0.934i·3-s − 0.999·5-s + 0.700i·7-s + 0.127·9-s + 1.69·11-s − 0.726i·13-s − 0.934i·15-s + 0.207i·17-s − 0.0334·19-s − 0.654·21-s + 0.208i·23-s + 1.00·25-s + 1.05i·27-s + 1.80·29-s + 0.385·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642900502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642900502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 - 1.61iT - 3T^{2} \) |
| 7 | \( 1 - 1.85iT - 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 + 2.61iT - 13T^{2} \) |
| 17 | \( 1 - 0.854iT - 17T^{2} \) |
| 19 | \( 1 + 0.145T + 19T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9.70iT - 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 1.70iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 5.23iT - 67T^{2} \) |
| 71 | \( 1 + 0.381T + 71T^{2} \) |
| 73 | \( 1 - 16.4iT - 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 - 7.70iT - 83T^{2} \) |
| 89 | \( 1 + 3.70T + 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−9.383226866345240309294557595151, −8.756681961364310166018026628000, −8.069319569926885834171896976353, −7.04226297846192532581681093713, −6.28413194147013180876171839497, −5.21023822604839441202975886528, −4.34425528069808022455527823395, −3.78083280748645323801804237318, −2.85545714268729118782634492363, −1.16496266223136356532337259221,
0.77518004565332744831613777496, 1.64866083417210551987370985058, 3.15955995242963240899943542471, 4.16650434119763488993331408300, 4.62806068679460302769009863625, 6.29720852306682266490098189385, 6.86019507879760787050634968953, 7.22715723877195782403349635457, 8.263551319419712130131247976950, 8.784385167738227097384581803497
