| L(s) = 1 | − 0.944·2-s + 7.13·3-s − 7.10·4-s − 5·5-s − 6.74·6-s + 21.6·7-s + 14.2·8-s + 23.9·9-s + 4.72·10-s − 55.2·11-s − 50.7·12-s + 24.7·13-s − 20.4·14-s − 35.6·15-s + 43.3·16-s − 22.6·18-s + 101.·19-s + 35.5·20-s + 154.·21-s + 52.1·22-s − 109.·23-s + 101.·24-s + 25·25-s − 23.3·26-s − 21.6·27-s − 153.·28-s + 0.675·29-s + ⋯ |
| L(s) = 1 | − 0.334·2-s + 1.37·3-s − 0.888·4-s − 0.447·5-s − 0.458·6-s + 1.16·7-s + 0.630·8-s + 0.887·9-s + 0.149·10-s − 1.51·11-s − 1.22·12-s + 0.527·13-s − 0.390·14-s − 0.614·15-s + 0.677·16-s − 0.296·18-s + 1.22·19-s + 0.397·20-s + 1.60·21-s + 0.505·22-s − 0.991·23-s + 0.866·24-s + 0.200·25-s − 0.176·26-s − 0.154·27-s − 1.03·28-s + 0.00432·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.370834159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370834159\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.944T + 8T^{2} \) |
| 3 | \( 1 - 7.13T + 27T^{2} \) |
| 7 | \( 1 - 21.6T + 343T^{2} \) |
| 11 | \( 1 + 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 0.675T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 118.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 741.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 187.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 838.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 95.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 743.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 520.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 963.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 969.T + 9.12e5T^{2} \) |
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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
−8.857652223719906128586193397815, −8.257938266072819336868675840662, −7.978282948654712695632550226828, −7.31208602048067169993526242804, −5.58527684055799827844728879509, −4.83249114137601125677336081964, −3.96710931737245562281928613709, −3.05301811769733971145541101856, −1.99397435264505536605033633970, −0.76501060244635937137648952807,
0.76501060244635937137648952807, 1.99397435264505536605033633970, 3.05301811769733971145541101856, 3.96710931737245562281928613709, 4.83249114137601125677336081964, 5.58527684055799827844728879509, 7.31208602048067169993526242804, 7.978282948654712695632550226828, 8.257938266072819336868675840662, 8.857652223719906128586193397815
