| L(s) = 1 | + 3.94·2-s + 7.57·4-s − 3.12·5-s + 13.3·7-s − 1.68·8-s − 12.3·10-s + 20.7·11-s + 18.2·13-s + 52.4·14-s − 67.2·16-s + 44.9·17-s + 18.3·19-s − 23.6·20-s + 81.9·22-s + 97.9·23-s − 115.·25-s + 71.8·26-s + 100.·28-s + 165.·29-s − 30.7·31-s − 251.·32-s + 177.·34-s − 41.5·35-s − 207.·37-s + 72.4·38-s + 5.25·40-s + 206.·41-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.946·4-s − 0.279·5-s + 0.718·7-s − 0.0742·8-s − 0.389·10-s + 0.568·11-s + 0.388·13-s + 1.00·14-s − 1.05·16-s + 0.641·17-s + 0.221·19-s − 0.264·20-s + 0.793·22-s + 0.888·23-s − 0.921·25-s + 0.541·26-s + 0.679·28-s + 1.05·29-s − 0.178·31-s − 1.39·32-s + 0.894·34-s − 0.200·35-s − 0.919·37-s + 0.309·38-s + 0.0207·40-s + 0.788·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.350785729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.350785729\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 3.94T + 8T^{2} \) |
| 5 | \( 1 + 3.12T + 125T^{2} \) |
| 7 | \( 1 - 13.3T + 343T^{2} \) |
| 11 | \( 1 - 20.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 18.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 30.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 308.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 372.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 323.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 355.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 702.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 137.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 233.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 420.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.76e3T + 9.12e5T^{2} \) |
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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
−8.653827341722677145409175570166, −7.83007464114579911396310301680, −6.94857407526092000275390721520, −6.16686021532273515149941424895, −5.36086698802916573955838018139, −4.70760280476374819221969184338, −3.88001977752895762934550292713, −3.22296582148206057730140342867, −2.08650434566604334577615553349, −0.872191629011791196579293251197,
0.872191629011791196579293251197, 2.08650434566604334577615553349, 3.22296582148206057730140342867, 3.88001977752895762934550292713, 4.70760280476374819221969184338, 5.36086698802916573955838018139, 6.16686021532273515149941424895, 6.94857407526092000275390721520, 7.83007464114579911396310301680, 8.653827341722677145409175570166
