| L(s) = 1 | + 2·2-s + 4·4-s + 19.6·5-s + 11.5·7-s + 8·8-s + 39.3·10-s + 37.6·11-s − 44.5·13-s + 23.1·14-s + 16·16-s + 61.1·17-s + 63.7·19-s + 78.7·20-s + 75.3·22-s − 177.·23-s + 262.·25-s − 89.1·26-s + 46.3·28-s − 29·29-s − 233.·31-s + 32·32-s + 122.·34-s + 228.·35-s + 10.2·37-s + 127.·38-s + 157.·40-s − 347.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.76·5-s + 0.626·7-s + 0.353·8-s + 1.24·10-s + 1.03·11-s − 0.951·13-s + 0.442·14-s + 0.250·16-s + 0.873·17-s + 0.770·19-s + 0.880·20-s + 0.729·22-s − 1.60·23-s + 2.10·25-s − 0.672·26-s + 0.313·28-s − 0.185·29-s − 1.35·31-s + 0.176·32-s + 0.617·34-s + 1.10·35-s + 0.0453·37-s + 0.544·38-s + 0.622·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.739200646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.739200646\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 5 | \( 1 - 19.6T + 125T^{2} \) |
| 7 | \( 1 - 11.5T + 343T^{2} \) |
| 11 | \( 1 - 37.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 233.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 14.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 606.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 702.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 543.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 314.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 859.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 725.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 820.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 60.9T + 9.12e5T^{2} \) |
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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
−10.21701042573357641930399154905, −9.846111260992404194210434381536, −8.830498439365485460829795364675, −7.53147800788109200435683751704, −6.56331672267320718109140106279, −5.62607358368003630103294246496, −5.06975710941976424177311316219, −3.68124620753667181528038513731, −2.25511307933427229245861535239, −1.45835195147610686305167528359,
1.45835195147610686305167528359, 2.25511307933427229245861535239, 3.68124620753667181528038513731, 5.06975710941976424177311316219, 5.62607358368003630103294246496, 6.56331672267320718109140106279, 7.53147800788109200435683751704, 8.830498439365485460829795364675, 9.846111260992404194210434381536, 10.21701042573357641930399154905
