| L(s) = 1 | − 1.13·2-s − 0.710·4-s − 5-s − 4.10·7-s + 3.07·8-s + 1.13·10-s + 2.27·13-s + 4.65·14-s − 2.07·16-s − 5.01·17-s + 7.57·19-s + 0.710·20-s + 1.97·23-s + 25-s − 2.58·26-s + 2.91·28-s + 4.00·29-s − 3.81·31-s − 3.79·32-s + 5.69·34-s + 4.10·35-s − 6.73·37-s − 8.60·38-s − 3.07·40-s + 4.02·41-s − 9.57·43-s − 2.24·46-s + ⋯ |
| L(s) = 1 | − 0.803·2-s − 0.355·4-s − 0.447·5-s − 1.55·7-s + 1.08·8-s + 0.359·10-s + 0.630·13-s + 1.24·14-s − 0.518·16-s − 1.21·17-s + 1.73·19-s + 0.158·20-s + 0.411·23-s + 0.200·25-s − 0.506·26-s + 0.550·28-s + 0.743·29-s − 0.685·31-s − 0.671·32-s + 0.977·34-s + 0.693·35-s − 1.10·37-s − 1.39·38-s − 0.486·40-s + 0.628·41-s − 1.46·43-s − 0.330·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4895639435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4895639435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 - 7.57T + 19T^{2} \) |
| 23 | \( 1 - 1.97T + 23T^{2} \) |
| 29 | \( 1 - 4.00T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 4.02T + 41T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 + 6.31T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 3.89T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−8.393183334022083662842040713090, −7.41882273126286939716105970278, −6.95398388999411867355474852898, −6.21131554887945167191633809074, −5.25311318983951563535471525138, −4.44567126825667221189431775805, −3.53160883088482252955460614442, −3.00866178474279022350039439676, −1.55836842660457048987955238742, −0.43899025539473538617771511375,
0.43899025539473538617771511375, 1.55836842660457048987955238742, 3.00866178474279022350039439676, 3.53160883088482252955460614442, 4.44567126825667221189431775805, 5.25311318983951563535471525138, 6.21131554887945167191633809074, 6.95398388999411867355474852898, 7.41882273126286939716105970278, 8.393183334022083662842040713090
