| L(s) = 1 | + 14.4·5-s + 35.5·7-s + 19.5·11-s + 62.5·13-s + 17·17-s + 78.5·19-s − 66.6·23-s + 84.7·25-s + 280.·29-s + 179.·31-s + 514.·35-s − 381.·37-s − 74.0·41-s − 22.2·43-s − 498.·47-s + 920.·49-s − 36.4·53-s + 282.·55-s − 748.·59-s − 730.·61-s + 906.·65-s − 781.·67-s + 674.·71-s − 906.·73-s + 694.·77-s + 1.07e3·79-s − 1.08e3·83-s + ⋯ |
| L(s) = 1 | + 1.29·5-s + 1.91·7-s + 0.535·11-s + 1.33·13-s + 0.242·17-s + 0.948·19-s − 0.604·23-s + 0.677·25-s + 1.79·29-s + 1.04·31-s + 2.48·35-s − 1.69·37-s − 0.281·41-s − 0.0787·43-s − 1.54·47-s + 2.68·49-s − 0.0945·53-s + 0.693·55-s − 1.65·59-s − 1.53·61-s + 1.72·65-s − 1.42·67-s + 1.12·71-s − 1.45·73-s + 1.02·77-s + 1.52·79-s − 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.290240557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.290240557\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 5 | \( 1 - 14.4T + 125T^{2} \) |
| 7 | \( 1 - 35.5T + 343T^{2} \) |
| 11 | \( 1 - 19.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 78.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 280.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 74.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 22.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 498.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 36.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 748.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 730.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 781.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 906.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 964.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 607.T + 9.12e5T^{2} \) |
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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
−9.297092412736631486662195382949, −8.471539278631277743344947857664, −7.964735662564576066891385217255, −6.72030125624680212487078237032, −5.93283137369673671258501672171, −5.14812842646043215008605556516, −4.36451248665820205300744526274, −3.00752112668854588758544391002, −1.58232496752584933811990733923, −1.35341730592656442427451010955,
1.35341730592656442427451010955, 1.58232496752584933811990733923, 3.00752112668854588758544391002, 4.36451248665820205300744526274, 5.14812842646043215008605556516, 5.93283137369673671258501672171, 6.72030125624680212487078237032, 7.964735662564576066891385217255, 8.471539278631277743344947857664, 9.297092412736631486662195382949
