| L(s) = 1 | − 5·5-s − 10.4·7-s + 33.3·11-s − 23.9·13-s − 72.5·17-s − 45.9·19-s − 163.·23-s + 25·25-s − 30.9·29-s + 304.·31-s + 52.4·35-s + 150.·37-s + 409.·41-s − 249.·43-s + 155.·47-s − 233.·49-s − 263.·53-s − 166.·55-s + 245.·59-s − 466.·61-s + 119.·65-s − 177.·67-s − 45.2·71-s − 949.·73-s − 349.·77-s + 496.·79-s − 354.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.565·7-s + 0.913·11-s − 0.510·13-s − 1.03·17-s − 0.555·19-s − 1.47·23-s + 0.200·25-s − 0.198·29-s + 1.76·31-s + 0.253·35-s + 0.667·37-s + 1.55·41-s − 0.883·43-s + 0.483·47-s − 0.679·49-s − 0.681·53-s − 0.408·55-s + 0.542·59-s − 0.979·61-s + 0.228·65-s − 0.322·67-s − 0.0755·71-s − 1.52·73-s − 0.517·77-s + 0.707·79-s − 0.468·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.290816485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290816485\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 10.4T + 343T^{2} \) |
| 11 | \( 1 - 33.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 30.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 245.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 177.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 45.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 949.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 496.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 354.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 408.T + 9.12e5T^{2} \) |
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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
−9.055356718068312149742291520433, −8.250686483204448209752112197481, −7.45039824810857017071487002544, −6.47281729104291634897038572007, −6.08262228253038984712305396696, −4.60475997368134007488671670644, −4.12282446076675879416946561903, −3.00876518651320113695130015563, −1.96056593115746893903930804758, −0.53072958907659564416184735706,
0.53072958907659564416184735706, 1.96056593115746893903930804758, 3.00876518651320113695130015563, 4.12282446076675879416946561903, 4.60475997368134007488671670644, 6.08262228253038984712305396696, 6.47281729104291634897038572007, 7.45039824810857017071487002544, 8.250686483204448209752112197481, 9.055356718068312149742291520433
