| L(s) = 1 | − 0.0516·2-s + 1.85·3-s − 1.99·4-s + 0.443·5-s − 0.0958·6-s − 3.87·7-s + 0.206·8-s + 0.452·9-s − 0.0228·10-s + 2.04·11-s − 3.71·12-s − 4.44·13-s + 0.199·14-s + 0.823·15-s + 3.98·16-s − 5.43·17-s − 0.0233·18-s − 4.61·19-s − 0.884·20-s − 7.19·21-s − 0.105·22-s + 3.00·23-s + 0.383·24-s − 4.80·25-s + 0.229·26-s − 4.73·27-s + 7.73·28-s + ⋯ |
| L(s) = 1 | − 0.0364·2-s + 1.07·3-s − 0.998·4-s + 0.198·5-s − 0.0391·6-s − 1.46·7-s + 0.0729·8-s + 0.150·9-s − 0.00723·10-s + 0.615·11-s − 1.07·12-s − 1.23·13-s + 0.0534·14-s + 0.212·15-s + 0.996·16-s − 1.31·17-s − 0.00550·18-s − 1.05·19-s − 0.197·20-s − 1.57·21-s − 0.0224·22-s + 0.627·23-s + 0.0782·24-s − 0.960·25-s + 0.0449·26-s − 0.910·27-s + 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9688143206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9688143206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0516T + 2T^{2} \) |
| 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 0.443T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 2.00T + 41T^{2} \) |
| 43 | \( 1 - 4.26T + 43T^{2} \) |
| 47 | \( 1 - 0.676T + 47T^{2} \) |
| 53 | \( 1 - 9.76T + 53T^{2} \) |
| 59 | \( 1 + 0.0603T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 - 0.421T + 67T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.056352586660072154361722695755, −7.10466052558123118805302086324, −6.57760761418714465535238772065, −5.80720186635184358839433012303, −4.84708467476541660344561982276, −4.12686523958023213301091151183, −3.52417754630596685475353007997, −2.71823023097503578768855264743, −2.05528192887927775008455431713, −0.43778257619022722776170640284,
0.43778257619022722776170640284, 2.05528192887927775008455431713, 2.71823023097503578768855264743, 3.52417754630596685475353007997, 4.12686523958023213301091151183, 4.84708467476541660344561982276, 5.80720186635184358839433012303, 6.57760761418714465535238772065, 7.10466052558123118805302086324, 8.056352586660072154361722695755
