| L(s) = 1 | + 0.538·2-s − 1.71·4-s + 1.52·5-s + 0.425·7-s − 1.99·8-s + 0.822·10-s − 5.10·11-s + 3.06·13-s + 0.229·14-s + 2.34·16-s + 3.41·17-s + 6.12·19-s − 2.61·20-s − 2.75·22-s − 7.29·23-s − 2.66·25-s + 1.65·26-s − 0.728·28-s − 4.75·29-s + 10.7·31-s + 5.25·32-s + 1.84·34-s + 0.650·35-s + 10.5·37-s + 3.29·38-s − 3.05·40-s − 9.59·41-s + ⋯ |
| L(s) = 1 | + 0.380·2-s − 0.855·4-s + 0.683·5-s + 0.161·7-s − 0.706·8-s + 0.260·10-s − 1.54·11-s + 0.850·13-s + 0.0613·14-s + 0.586·16-s + 0.828·17-s + 1.40·19-s − 0.584·20-s − 0.586·22-s − 1.52·23-s − 0.533·25-s + 0.323·26-s − 0.137·28-s − 0.882·29-s + 1.93·31-s + 0.929·32-s + 0.315·34-s + 0.110·35-s + 1.73·37-s + 0.534·38-s − 0.482·40-s − 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936148580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936148580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 0.538T + 2T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 - 0.425T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 6.12T + 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 9.59T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 9.36T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 7.81T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 6.55T + 71T^{2} \) |
| 73 | \( 1 + 0.977T + 73T^{2} \) |
| 79 | \( 1 - 6.78T + 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 9.78T + 97T^{2} \) |
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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
−8.001972436907586348168390494372, −7.68217091593840233152129141570, −6.32723478285593759241677367297, −5.78185555297150172420824914061, −5.26604321034125226104094175033, −4.59137330592159607859517486696, −3.59394261056375248181050894185, −2.97547555457837756238533680391, −1.88831051891260826359446546082, −0.69666363228402572258154867937,
0.69666363228402572258154867937, 1.88831051891260826359446546082, 2.97547555457837756238533680391, 3.59394261056375248181050894185, 4.59137330592159607859517486696, 5.26604321034125226104094175033, 5.78185555297150172420824914061, 6.32723478285593759241677367297, 7.68217091593840233152129141570, 8.001972436907586348168390494372
