| L(s) = 1 | + 9·3-s − 67.5·5-s + 95.5·7-s + 81·9-s + 358.·11-s − 169·13-s − 608.·15-s − 1.90e3·17-s + 1.57e3·19-s + 860.·21-s + 1.51e3·23-s + 1.44e3·25-s + 729·27-s − 2.98e3·29-s + 3.79e3·31-s + 3.22e3·33-s − 6.46e3·35-s + 4.46e3·37-s − 1.52e3·39-s + 5.28e3·41-s − 1.01e4·43-s − 5.47e3·45-s + 6.97e3·47-s − 7.67e3·49-s − 1.71e4·51-s − 9.15e3·53-s − 2.42e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.20·5-s + 0.737·7-s + 0.333·9-s + 0.893·11-s − 0.277·13-s − 0.698·15-s − 1.60·17-s + 1.00·19-s + 0.425·21-s + 0.598·23-s + 0.461·25-s + 0.192·27-s − 0.658·29-s + 0.709·31-s + 0.516·33-s − 0.891·35-s + 0.535·37-s − 0.160·39-s + 0.490·41-s − 0.837·43-s − 0.402·45-s + 0.460·47-s − 0.456·49-s − 0.924·51-s − 0.447·53-s − 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.306118587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.306118587\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 + 67.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 95.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 358.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.90e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.28e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.15e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.55e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 528.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.23e5T + 8.58e9T^{2} \) |
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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
−9.608492558131173050674859010110, −8.876938081505045101186637111331, −8.057117528303973454014846336664, −7.37752855187079046669474253676, −6.49304330949271432067933704805, −4.92232796835032584511801343552, −4.20563994946450867803776276154, −3.28840277876028853515307518667, −1.99280869206188197649617503455, −0.71781670754388292509919460849,
0.71781670754388292509919460849, 1.99280869206188197649617503455, 3.28840277876028853515307518667, 4.20563994946450867803776276154, 4.92232796835032584511801343552, 6.49304330949271432067933704805, 7.37752855187079046669474253676, 8.057117528303973454014846336664, 8.876938081505045101186637111331, 9.608492558131173050674859010110
