| L(s) = 1 | + 2.54·2-s + 3-s + 4.48·4-s + 0.532·5-s + 2.54·6-s + 0.266·7-s + 6.32·8-s + 9-s + 1.35·10-s + 3.13·11-s + 4.48·12-s + 4.93·13-s + 0.677·14-s + 0.532·15-s + 7.13·16-s + 17-s + 2.54·18-s − 4.75·19-s + 2.38·20-s + 0.266·21-s + 7.98·22-s − 4.73·23-s + 6.32·24-s − 4.71·25-s + 12.5·26-s + 27-s + 1.19·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.577·3-s + 2.24·4-s + 0.238·5-s + 1.03·6-s + 0.100·7-s + 2.23·8-s + 0.333·9-s + 0.428·10-s + 0.946·11-s + 1.29·12-s + 1.36·13-s + 0.181·14-s + 0.137·15-s + 1.78·16-s + 0.242·17-s + 0.600·18-s − 1.09·19-s + 0.533·20-s + 0.0580·21-s + 1.70·22-s − 0.988·23-s + 1.29·24-s − 0.943·25-s + 2.46·26-s + 0.192·27-s + 0.225·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.510938743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.510938743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 5 | \( 1 - 0.532T + 5T^{2} \) |
| 7 | \( 1 - 0.266T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 3.64T + 47T^{2} \) |
| 53 | \( 1 + 8.29T + 53T^{2} \) |
| 59 | \( 1 - 7.56T + 59T^{2} \) |
| 61 | \( 1 + 2.77T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 + 6.65T + 71T^{2} \) |
| 73 | \( 1 - 1.86T + 73T^{2} \) |
| 83 | \( 1 - 7.57T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 7.15T + 97T^{2} \) |
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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
−8.264708255278791755581675830086, −7.56245496559848537523652589629, −6.50976633957829567598739962188, −6.20504769753552515166896357146, −5.49178660617728990807803717101, −4.33666317669393125068677428606, −3.97357859391364176123722271116, −3.28856726841288010255820093675, −2.22250012929468804553102722357, −1.51046941928847910118595587737,
1.51046941928847910118595587737, 2.22250012929468804553102722357, 3.28856726841288010255820093675, 3.97357859391364176123722271116, 4.33666317669393125068677428606, 5.49178660617728990807803717101, 6.20504769753552515166896357146, 6.50976633957829567598739962188, 7.56245496559848537523652589629, 8.264708255278791755581675830086
