| L(s) = 1 | + 2.23·2-s + 3.00·4-s − 2·5-s + 4.47·7-s + 2.23·8-s − 4.47·10-s + 10.0·14-s − 0.999·16-s + 4.47·17-s + 4.47·19-s − 6.00·20-s + 4·23-s − 25-s + 13.4·28-s + 4.47·29-s − 6.70·32-s + 10.0·34-s − 8.94·35-s + 2·37-s + 10.0·38-s − 4.47·40-s − 4.47·41-s − 4.47·43-s + 8.94·46-s − 8·47-s + 13.0·49-s − 2.23·50-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s − 0.894·5-s + 1.69·7-s + 0.790·8-s − 1.41·10-s + 2.67·14-s − 0.249·16-s + 1.08·17-s + 1.02·19-s − 1.34·20-s + 0.834·23-s − 0.200·25-s + 2.53·28-s + 0.830·29-s − 1.18·32-s + 1.71·34-s − 1.51·35-s + 0.328·37-s + 1.62·38-s − 0.707·40-s − 0.698·41-s − 0.681·43-s + 1.31·46-s − 1.16·47-s + 1.85·49-s − 0.316·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.020971377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.020971377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.15508704915123558559316011715, −8.830071836324234856711327103377, −7.86976955405133154697968732802, −7.42223965484766108483887071635, −6.23938317232482195788363899627, −5.08983680960020632100950710119, −4.85437285232922505847135247610, −3.78974801390572590287794033981, −2.95751143962211203424475359449, −1.46793275927465469289389877459,
1.46793275927465469289389877459, 2.95751143962211203424475359449, 3.78974801390572590287794033981, 4.85437285232922505847135247610, 5.08983680960020632100950710119, 6.23938317232482195788363899627, 7.42223965484766108483887071635, 7.86976955405133154697968732802, 8.830071836324234856711327103377, 10.15508704915123558559316011715
