| L(s) = 1 | + 2-s − 0.783·3-s + 4-s + 4.00·5-s − 0.783·6-s + 0.255·7-s + 8-s − 2.38·9-s + 4.00·10-s − 1.94·11-s − 0.783·12-s + 0.466·13-s + 0.255·14-s − 3.13·15-s + 16-s − 1.68·17-s − 2.38·18-s + 0.0117·19-s + 4.00·20-s − 0.200·21-s − 1.94·22-s − 23-s − 0.783·24-s + 11.0·25-s + 0.466·26-s + 4.22·27-s + 0.255·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.452·3-s + 0.5·4-s + 1.79·5-s − 0.320·6-s + 0.0964·7-s + 0.353·8-s − 0.795·9-s + 1.26·10-s − 0.585·11-s − 0.226·12-s + 0.129·13-s + 0.0681·14-s − 0.810·15-s + 0.250·16-s − 0.409·17-s − 0.562·18-s + 0.00270·19-s + 0.895·20-s − 0.0436·21-s − 0.414·22-s − 0.208·23-s − 0.160·24-s + 2.20·25-s + 0.0914·26-s + 0.812·27-s + 0.0482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.678475668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.678475668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.783T + 3T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 7 | \( 1 - 0.255T + 7T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 13 | \( 1 - 0.466T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 - 0.0117T + 19T^{2} \) |
| 29 | \( 1 - 9.98T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 1.60T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 7.24T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 0.104T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 0.362T + 89T^{2} \) |
| 97 | \( 1 - 6.04T + 97T^{2} \) |
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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
−8.162675405066039544526394030694, −6.94358197184125427988104639415, −6.47558509889527447661888347254, −5.80693097286756079099910344059, −5.29110303706423486735693536358, −4.82441260029667695790588646589, −3.60869259341633223018894154270, −2.54649082683997947875899824475, −2.19965014060206509303986764663, −0.932632992839163366382270777028,
0.932632992839163366382270777028, 2.19965014060206509303986764663, 2.54649082683997947875899824475, 3.60869259341633223018894154270, 4.82441260029667695790588646589, 5.29110303706423486735693536358, 5.80693097286756079099910344059, 6.47558509889527447661888347254, 6.94358197184125427988104639415, 8.162675405066039544526394030694
