L(s) = 1 | − 2-s + 4-s + 2.58·5-s + 0.635·7-s − 8-s − 2.58·10-s − 3.51·11-s + 5.12·13-s − 0.635·14-s + 16-s + 0.322·17-s + 3.38·19-s + 2.58·20-s + 3.51·22-s − 3.77·23-s + 1.68·25-s − 5.12·26-s + 0.635·28-s − 2.46·29-s + 1.98·31-s − 32-s − 0.322·34-s + 1.64·35-s − 7.77·37-s − 3.38·38-s − 2.58·40-s − 10.7·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.15·5-s + 0.240·7-s − 0.353·8-s − 0.817·10-s − 1.05·11-s + 1.42·13-s − 0.169·14-s + 0.250·16-s + 0.0781·17-s + 0.776·19-s + 0.578·20-s + 0.748·22-s − 0.787·23-s + 0.337·25-s − 1.00·26-s + 0.120·28-s − 0.457·29-s + 0.357·31-s − 0.176·32-s − 0.0552·34-s + 0.277·35-s − 1.27·37-s − 0.548·38-s − 0.408·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 - 0.635T + 7T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 0.322T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 23 | \( 1 + 3.77T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 1.98T + 31T^{2} \) |
| 37 | \( 1 + 7.77T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 9.96T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 0.586T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 0.0667T + 71T^{2} \) |
| 73 | \( 1 + 2.81T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.57314687526102423174281364247, −6.61111802397787090611605740593, −6.12396453492350955858894217950, −5.44340896642139903735743595232, −4.86659882112784040273185182217, −3.57333890460361098376691576699, −2.95650224472238825723628416062, −1.82921007331480556738559488208, −1.48203886759152582172895684574, 0,
1.48203886759152582172895684574, 1.82921007331480556738559488208, 2.95650224472238825723628416062, 3.57333890460361098376691576699, 4.86659882112784040273185182217, 5.44340896642139903735743595232, 6.12396453492350955858894217950, 6.61111802397787090611605740593, 7.57314687526102423174281364247