| L(s) = 1 | − 1.24·3-s − 2.66·5-s + 0.381·7-s − 1.44·9-s − 2.03·11-s − 3.34·13-s + 3.31·15-s − 0.943·17-s + 2.79·19-s − 0.476·21-s + 4.16·23-s + 2.07·25-s + 5.54·27-s − 7.67·29-s − 4.12·31-s + 2.53·33-s − 1.01·35-s − 3.07·37-s + 4.17·39-s − 7.88·41-s + 3.84·45-s − 4.12·47-s − 6.85·49-s + 1.17·51-s − 12.2·53-s + 5.40·55-s − 3.48·57-s + ⋯ |
| L(s) = 1 | − 0.719·3-s − 1.18·5-s + 0.144·7-s − 0.481·9-s − 0.612·11-s − 0.928·13-s + 0.856·15-s − 0.228·17-s + 0.641·19-s − 0.103·21-s + 0.867·23-s + 0.415·25-s + 1.06·27-s − 1.42·29-s − 0.740·31-s + 0.440·33-s − 0.171·35-s − 0.504·37-s + 0.668·39-s − 1.23·41-s + 0.573·45-s − 0.601·47-s − 0.979·49-s + 0.164·51-s − 1.67·53-s + 0.728·55-s − 0.461·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2030419005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2030419005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 \) |
| good | 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 + 0.943T + 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 + 7.67T + 29T^{2} \) |
| 31 | \( 1 + 4.12T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 + 7.88T + 41T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.40T + 59T^{2} \) |
| 61 | \( 1 + 6.72T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 + 6.09T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 0.0643T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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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
−7.79463817157471543520244552722, −7.27403335536516596140348182812, −6.62206391419092601658606839549, −5.60480730150095741811773864133, −5.11014013455936822311388919276, −4.51915929541580164017377487276, −3.47635863120916986086850353938, −2.91521893106317410548038633410, −1.68264159415108759694014625616, −0.22847697499404655061760893820,
0.22847697499404655061760893820, 1.68264159415108759694014625616, 2.91521893106317410548038633410, 3.47635863120916986086850353938, 4.51915929541580164017377487276, 5.11014013455936822311388919276, 5.60480730150095741811773864133, 6.62206391419092601658606839549, 7.27403335536516596140348182812, 7.79463817157471543520244552722
