| L(s) = 1 | + 5·5-s − 23.9·7-s + 57.9·11-s − 8.16·13-s + 50.0·17-s − 69.7·19-s + 4.92·23-s + 25·25-s − 79.4·29-s − 260.·31-s − 119.·35-s − 223.·37-s + 337.·41-s − 326.·43-s + 89.6·47-s + 229.·49-s + 543.·53-s + 289.·55-s + 92·59-s + 159.·61-s − 40.8·65-s + 910.·67-s − 293.·71-s + 142.·73-s − 1.38e3·77-s − 1.10e3·79-s + 813.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.29·7-s + 1.58·11-s − 0.174·13-s + 0.714·17-s − 0.842·19-s + 0.0446·23-s + 0.200·25-s − 0.508·29-s − 1.50·31-s − 0.577·35-s − 0.994·37-s + 1.28·41-s − 1.15·43-s + 0.278·47-s + 0.668·49-s + 1.40·53-s + 0.710·55-s + 0.203·59-s + 0.334·61-s − 0.0778·65-s + 1.66·67-s − 0.490·71-s + 0.227·73-s − 2.05·77-s − 1.57·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 - 57.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.16T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.92T + 1.21e4T^{2} \) |
| 29 | \( 1 + 79.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92T + 2.05e5T^{2} \) |
| 61 | \( 1 - 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 910.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 293.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 956.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 106.T + 9.12e5T^{2} \) |
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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
−8.586073635541684526636636617397, −7.31858737809018870030372597318, −6.72296317108167638162863543011, −6.07736387890552511899348628238, −5.30887165113565709222079893066, −3.97956198606435513412392390498, −3.51203911652305220053848828474, −2.32885282283661426561726151817, −1.25664347571108240023114863116, 0,
1.25664347571108240023114863116, 2.32885282283661426561726151817, 3.51203911652305220053848828474, 3.97956198606435513412392390498, 5.30887165113565709222079893066, 6.07736387890552511899348628238, 6.72296317108167638162863543011, 7.31858737809018870030372597318, 8.586073635541684526636636617397
