| L(s) = 1 | − 2.14·2-s + 2.01·3-s + 2.59·4-s + 5-s − 4.31·6-s − 3.47·7-s − 1.27·8-s + 1.04·9-s − 2.14·10-s + 11-s + 5.21·12-s + 6.08·13-s + 7.44·14-s + 2.01·15-s − 2.45·16-s − 0.860·17-s − 2.24·18-s + 5.32·19-s + 2.59·20-s − 6.98·21-s − 2.14·22-s + 5.95·23-s − 2.56·24-s + 25-s − 13.0·26-s − 3.93·27-s − 9.01·28-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.16·3-s + 1.29·4-s + 0.447·5-s − 1.76·6-s − 1.31·7-s − 0.450·8-s + 0.348·9-s − 0.677·10-s + 0.301·11-s + 1.50·12-s + 1.68·13-s + 1.99·14-s + 0.519·15-s − 0.614·16-s − 0.208·17-s − 0.528·18-s + 1.22·19-s + 0.580·20-s − 1.52·21-s − 0.456·22-s + 1.24·23-s − 0.523·24-s + 0.200·25-s − 2.55·26-s − 0.756·27-s − 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.416709650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416709650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 3 | \( 1 - 2.01T + 3T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 13 | \( 1 - 6.08T + 13T^{2} \) |
| 17 | \( 1 + 0.860T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 - 5.45T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 + 9.39T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 - 9.98T + 71T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 0.440T + 97T^{2} \) |
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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
−8.747448488431643660327226079973, −7.893226810538196103921551708157, −7.32840060706668916262203741302, −6.40102908686957862579901324532, −5.97402518039202897532671968734, −4.48376788963474716814936945618, −3.21624773277632217025073916164, −3.01795833477426070235856243472, −1.73793458717656159253314772533, −0.845783719463800036104152932469,
0.845783719463800036104152932469, 1.73793458717656159253314772533, 3.01795833477426070235856243472, 3.21624773277632217025073916164, 4.48376788963474716814936945618, 5.97402518039202897532671968734, 6.40102908686957862579901324532, 7.32840060706668916262203741302, 7.893226810538196103921551708157, 8.747448488431643660327226079973
