| L(s) = 1 | − i·2-s − 4-s + (−1.74 + 1.40i)5-s − 3.20i·7-s + i·8-s + (1.40 + 1.74i)10-s − 3.82·11-s − 0.147i·13-s − 3.20·14-s + 16-s − 0.978i·17-s − 2.67·19-s + (1.74 − 1.40i)20-s + 3.82i·22-s − 2.33i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.779 + 0.626i)5-s − 1.21i·7-s + 0.353i·8-s + (0.443 + 0.551i)10-s − 1.15·11-s − 0.0408i·13-s − 0.857·14-s + 0.250·16-s − 0.237i·17-s − 0.613·19-s + (0.389 − 0.313i)20-s + 0.815i·22-s − 0.487i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5825537630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5825537630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.74 - 1.40i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.20iT - 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 0.147iT - 13T^{2} \) |
| 17 | \( 1 + 0.978iT - 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 2.33iT - 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 + 3.62T + 31T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.53iT - 43T^{2} \) |
| 47 | \( 1 + 6.23iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 2.80iT - 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 13.9iT - 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 + 3.07iT - 97T^{2} \) |
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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
−8.382169326186111760636725473794, −8.222027944302948363864948654032, −7.12939480763848028487992761115, −6.80049178534065510686960758774, −5.49854168673241846906607368464, −4.58302487957843226182823705497, −3.96781127858357186579928994692, −3.12605426197819657557362811390, −2.34090463162509665388290559326, −0.852291516328369673074908161122,
0.23303016722920847646917732496, 1.90637882642582063835296350766, 3.04404434325576556119274430335, 3.99845564088502513390189800489, 5.09139570994424821017895799322, 5.28563260770443874148690315615, 6.27973958540223633904615723771, 7.12266003934201456084836765493, 7.977547067136750666304162322500, 8.461606680210381152369242811810
