| L(s) = 1 | + 0.585i·3-s − 4.82i·7-s + 2.65·9-s − 2·11-s − 2.24i·13-s − 4.82i·17-s + 19-s + 2.82·21-s + 6i·23-s + 3.31i·27-s − 10.4·29-s − 1.17·31-s − 1.17i·33-s − 10.2i·37-s + 1.31·39-s + ⋯ |
| L(s) = 1 | + 0.338i·3-s − 1.82i·7-s + 0.885·9-s − 0.603·11-s − 0.621i·13-s − 1.17i·17-s + 0.229·19-s + 0.617·21-s + 1.25i·23-s + 0.637i·27-s − 1.94·29-s − 0.210·31-s − 0.203i·33-s − 1.68i·37-s + 0.210·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.240746167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240746167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.585iT - 3T^{2} \) |
| 7 | \( 1 + 4.82iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 4.82iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 0.828iT - 43T^{2} \) |
| 47 | \( 1 - 0.828iT - 47T^{2} \) |
| 53 | \( 1 - 5.07iT - 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 0.585iT - 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 + 9.31iT - 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.75iT - 97T^{2} \) |
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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
−9.329281422350675336055303414628, −7.73755213979600958324637983219, −7.53538556137534499880034663050, −6.88378941129265558520382301297, −5.55695153622371043713285044332, −4.83156699223862316259037237170, −3.89838532985512108090909406237, −3.31064518014614791860158522328, −1.69876894568532541943955976164, −0.43965891675723827178753030623,
1.70306319486223363962787945059, 2.35724105000585277455318473274, 3.56433024233249576319646014572, 4.70788287172161161676475745778, 5.51093407788726407953362736484, 6.30208408258863143993010291043, 7.00113335969234800362936106053, 8.140698389547391115969866666685, 8.525248525222699455922657116350, 9.457567330720657875628495324974
