| L(s) = 1 | − 3.23·3-s + 7-s + 7.47·9-s + 0.236·11-s − 1.23·13-s − 2.47·17-s + 4.47·19-s − 3.23·21-s + 6.23·23-s − 14.4·27-s + 5·29-s − 3.70·31-s − 0.763·33-s − 3·37-s + 4.00·39-s + 4.76·41-s + 1.76·43-s − 2·47-s + 49-s + 8.00·51-s − 8.47·53-s − 14.4·57-s − 11.7·59-s − 9.70·61-s + 7.47·63-s − 4.23·67-s − 20.1·69-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 0.377·7-s + 2.49·9-s + 0.0711·11-s − 0.342·13-s − 0.599·17-s + 1.02·19-s − 0.706·21-s + 1.30·23-s − 2.78·27-s + 0.928·29-s − 0.666·31-s − 0.132·33-s − 0.493·37-s + 0.640·39-s + 0.744·41-s + 0.268·43-s − 0.291·47-s + 0.142·49-s + 1.12·51-s − 1.16·53-s − 1.91·57-s − 1.52·59-s − 1.24·61-s + 0.941·63-s − 0.517·67-s − 2.42·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9131947241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9131947241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 5.23T + 97T^{2} \) |
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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
−8.997817766239093368584896637960, −7.71636946585161733462585470812, −7.15871599984143801493225201712, −6.40248591351541448346145230388, −5.72578583867714300749631665858, −4.85763268688068697215320293662, −4.57977576161775762894777885259, −3.22105487340315837040061081294, −1.69974817186533828264479277417, −0.67762939405580346289441619077,
0.67762939405580346289441619077, 1.69974817186533828264479277417, 3.22105487340315837040061081294, 4.57977576161775762894777885259, 4.85763268688068697215320293662, 5.72578583867714300749631665858, 6.40248591351541448346145230388, 7.15871599984143801493225201712, 7.71636946585161733462585470812, 8.997817766239093368584896637960
