| L(s) = 1 | − 2i·3-s − 3.46i·5-s + 4.73·7-s − 9-s + 1.26i·11-s + i·13-s − 6.92·15-s − 1.46·17-s + 2.73i·19-s − 9.46i·21-s − 4·23-s − 6.99·25-s − 4i·27-s + 2i·29-s + 3.26·31-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 1.54i·5-s + 1.78·7-s − 0.333·9-s + 0.382i·11-s + 0.277i·13-s − 1.78·15-s − 0.355·17-s + 0.626i·19-s − 2.06i·21-s − 0.834·23-s − 1.39·25-s − 0.769i·27-s + 0.371i·29-s + 0.586·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.987792 - 1.28731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987792 - 1.28731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 1.26iT - 11T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 2.73iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 - 4.92iT - 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 - 7.46iT - 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + 0.196iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 2.73iT - 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 + 6.73iT - 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−11.31612240228864407876460987867, −9.972408185142455694800867205414, −8.744082584547826445155678671743, −8.131188689436848486045260496386, −7.53917559313879621791417810081, −6.19942722935809631447847378801, −4.98462993962920393861146330572, −4.38201926300298958803600849563, −1.91365119986622245651772896758, −1.25787348341261807773421410633,
2.22449239575256709356198648105, 3.57585239248970004918043916706, 4.56341897588010696309581687180, 5.55997228536087349195201798109, 6.86174036002332583599412977952, 7.83355508684672240742556567360, 8.758831745200813004232375446695, 9.998905142741516134238077626976, 10.64073027091680827405028239155, 11.18347939226068674769171128578
