| L(s) = 1 | − 2.26·2-s − 1.83·3-s + 3.12·4-s + 1.84·5-s + 4.15·6-s + 7-s − 2.55·8-s + 0.368·9-s − 4.17·10-s − 4.09·11-s − 5.73·12-s − 3.93·13-s − 2.26·14-s − 3.38·15-s − 0.477·16-s + 6.99·17-s − 0.834·18-s − 4.65·19-s + 5.76·20-s − 1.83·21-s + 9.26·22-s − 0.636·23-s + 4.68·24-s − 1.60·25-s + 8.91·26-s + 4.82·27-s + 3.12·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 1.05·3-s + 1.56·4-s + 0.824·5-s + 1.69·6-s + 0.377·7-s − 0.901·8-s + 0.122·9-s − 1.32·10-s − 1.23·11-s − 1.65·12-s − 1.09·13-s − 0.605·14-s − 0.873·15-s − 0.119·16-s + 1.69·17-s − 0.196·18-s − 1.06·19-s + 1.28·20-s − 0.400·21-s + 1.97·22-s − 0.132·23-s + 0.955·24-s − 0.320·25-s + 1.74·26-s + 0.929·27-s + 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2156229762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2156229762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 + 1.83T + 3T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 + 0.636T + 23T^{2} \) |
| 29 | \( 1 + 9.22T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 3.99T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 6.71T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.984227641680390089047010594052, −7.61010950314067943468174475898, −6.79746951443975462042133252875, −6.03563391990809649270164854413, −5.25731773013315695864236168493, −4.99242790183355318087052400123, −3.37549524440643263738718916920, −2.19284612555078635131793937485, −1.68553104726451898875068040771, −0.31926432379326416509613190554,
0.31926432379326416509613190554, 1.68553104726451898875068040771, 2.19284612555078635131793937485, 3.37549524440643263738718916920, 4.99242790183355318087052400123, 5.25731773013315695864236168493, 6.03563391990809649270164854413, 6.79746951443975462042133252875, 7.61010950314067943468174475898, 7.984227641680390089047010594052
