| L(s) = 1 | − 0.694·2-s − 31.5·4-s − 25·5-s + 83.1·7-s + 44.1·8-s + 17.3·10-s − 121·11-s − 674.·13-s − 57.7·14-s + 977.·16-s − 1.92e3·17-s − 149.·19-s + 787.·20-s + 84.0·22-s + 1.35e3·23-s + 625·25-s + 468.·26-s − 2.62e3·28-s + 7.32e3·29-s − 4.21e3·31-s − 2.09e3·32-s + 1.33e3·34-s − 2.07e3·35-s − 1.34e4·37-s + 104.·38-s − 1.10e3·40-s − 2.86e3·41-s + ⋯ |
| L(s) = 1 | − 0.122·2-s − 0.984·4-s − 0.447·5-s + 0.641·7-s + 0.243·8-s + 0.0549·10-s − 0.301·11-s − 1.10·13-s − 0.0787·14-s + 0.954·16-s − 1.61·17-s − 0.0951·19-s + 0.440·20-s + 0.0370·22-s + 0.534·23-s + 0.200·25-s + 0.135·26-s − 0.631·28-s + 1.61·29-s − 0.787·31-s − 0.361·32-s + 0.198·34-s − 0.286·35-s − 1.61·37-s + 0.0116·38-s − 0.109·40-s − 0.266·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7959918618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7959918618\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 0.694T + 32T^{2} \) |
| 7 | \( 1 - 83.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 674.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.33e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.65e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.37e4T + 8.58e9T^{2} \) |
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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
−10.16260908528484151646752069252, −9.086253849255547889045641472559, −8.469015215439844258900122145900, −7.60073584486148454846809951154, −6.59704470162356467193931530052, −4.91309294952406055387739865405, −4.78624471986705496006413912244, −3.41375113520436924607620786305, −1.98130498614868975564402576676, −0.45238787643996408916638723705,
0.45238787643996408916638723705, 1.98130498614868975564402576676, 3.41375113520436924607620786305, 4.78624471986705496006413912244, 4.91309294952406055387739865405, 6.59704470162356467193931530052, 7.60073584486148454846809951154, 8.469015215439844258900122145900, 9.086253849255547889045641472559, 10.16260908528484151646752069252
