| L(s) = 1 | − 4.69·3-s − 10.2·7-s − 220.·9-s − 596.·11-s − 420.·13-s + 974.·17-s + 380.·19-s + 48.1·21-s − 3.54e3·23-s + 2.17e3·27-s + 5.44e3·29-s + 3.62e3·31-s + 2.79e3·33-s + 1.75e3·37-s + 1.97e3·39-s + 263.·41-s − 1.44e4·43-s − 2.34e4·47-s − 1.67e4·49-s − 4.57e3·51-s + 3.34e4·53-s − 1.78e3·57-s + 2.90e3·59-s + 2.94e4·61-s + 2.26e3·63-s − 7.16e3·67-s + 1.66e4·69-s + ⋯ |
| L(s) = 1 | − 0.301·3-s − 0.0791·7-s − 0.909·9-s − 1.48·11-s − 0.690·13-s + 0.817·17-s + 0.241·19-s + 0.0238·21-s − 1.39·23-s + 0.574·27-s + 1.20·29-s + 0.677·31-s + 0.447·33-s + 0.210·37-s + 0.207·39-s + 0.0245·41-s − 1.18·43-s − 1.54·47-s − 0.993·49-s − 0.246·51-s + 1.63·53-s − 0.0728·57-s + 0.108·59-s + 1.01·61-s + 0.0719·63-s − 0.194·67-s + 0.420·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.006844555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006844555\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.69T + 243T^{2} \) |
| 7 | \( 1 + 10.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 596.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 420.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 974.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 380.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 263.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.44e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.34e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.16e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.29e4T + 8.58e9T^{2} \) |
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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
−10.28381808858882794235724169687, −9.867350238935155202726808823363, −8.355723263293293066721991989006, −7.889783888345293149066411542626, −6.59418631380264099003557391928, −5.55220867546014413241722825663, −4.85023867596827801048262894324, −3.26062358686159251309644044518, −2.29760839539688581443738404564, −0.51135943455962205677676753847,
0.51135943455962205677676753847, 2.29760839539688581443738404564, 3.26062358686159251309644044518, 4.85023867596827801048262894324, 5.55220867546014413241722825663, 6.59418631380264099003557391928, 7.889783888345293149066411542626, 8.355723263293293066721991989006, 9.867350238935155202726808823363, 10.28381808858882794235724169687
