| L(s) = 1 | + 2-s − 0.102·3-s + 4-s + 3.23·5-s − 0.102·6-s − 2.42·7-s + 8-s − 2.98·9-s + 3.23·10-s + 0.588·11-s − 0.102·12-s + 0.301·13-s − 2.42·14-s − 0.331·15-s + 16-s + 3.72·17-s − 2.98·18-s + 4.71·19-s + 3.23·20-s + 0.248·21-s + 0.588·22-s + 1.45·23-s − 0.102·24-s + 5.46·25-s + 0.301·26-s + 0.613·27-s − 2.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0591·3-s + 0.5·4-s + 1.44·5-s − 0.0418·6-s − 0.918·7-s + 0.353·8-s − 0.996·9-s + 1.02·10-s + 0.177·11-s − 0.0295·12-s + 0.0836·13-s − 0.649·14-s − 0.0855·15-s + 0.250·16-s + 0.904·17-s − 0.704·18-s + 1.08·19-s + 0.723·20-s + 0.0543·21-s + 0.125·22-s + 0.303·23-s − 0.0209·24-s + 1.09·25-s + 0.0591·26-s + 0.118·27-s − 0.459·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.601022269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.601022269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 0.102T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 - 0.588T + 11T^{2} \) |
| 13 | \( 1 - 0.301T + 13T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 - 1.45T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 - 4.67T + 43T^{2} \) |
| 47 | \( 1 + 8.27T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 3.38T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 6.06T + 73T^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 - 5.69T + 83T^{2} \) |
| 89 | \( 1 + 1.43T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.504206463042560003230507360044, −7.55898837464024004032982409072, −6.60380374374591572852853513829, −6.15405469358998288450435946687, −5.49193819859244732952492981131, −5.00872821498592756236551990693, −3.61734324650480653122879133680, −3.00609811249799154389911380815, −2.23659483704496017766744393845, −1.00982011728101447271722618252,
1.00982011728101447271722618252, 2.23659483704496017766744393845, 3.00609811249799154389911380815, 3.61734324650480653122879133680, 5.00872821498592756236551990693, 5.49193819859244732952492981131, 6.15405469358998288450435946687, 6.60380374374591572852853513829, 7.55898837464024004032982409072, 8.504206463042560003230507360044
