| L(s) = 1 | + 10.2·2-s + 72.1·4-s + 55.5·5-s + 2.50·7-s + 409.·8-s + 566.·10-s − 228.·11-s + 658.·13-s + 25.5·14-s + 1.87e3·16-s + 1.44e3·17-s − 982.·19-s + 4.00e3·20-s − 2.33e3·22-s − 529·23-s − 42.8·25-s + 6.72e3·26-s + 180.·28-s + 7.15e3·29-s − 9.25e3·31-s + 5.98e3·32-s + 1.47e4·34-s + 139.·35-s + 2.42e3·37-s − 1.00e4·38-s + 2.27e4·40-s + 4.07e3·41-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.25·4-s + 0.993·5-s + 0.0193·7-s + 2.26·8-s + 1.79·10-s − 0.569·11-s + 1.08·13-s + 0.0348·14-s + 1.82·16-s + 1.21·17-s − 0.624·19-s + 2.23·20-s − 1.02·22-s − 0.208·23-s − 0.0137·25-s + 1.95·26-s + 0.0435·28-s + 1.58·29-s − 1.73·31-s + 1.03·32-s + 2.18·34-s + 0.0191·35-s + 0.290·37-s − 1.12·38-s + 2.24·40-s + 0.378·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.395159475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.395159475\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 10.2T + 32T^{2} \) |
| 5 | \( 1 - 55.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 2.50T + 1.68e4T^{2} \) |
| 11 | \( 1 + 228.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 658.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 982.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.42e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.07e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.26e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.61e4T + 8.58e9T^{2} \) |
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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
−11.80715044539871697007724302341, −10.80859912131595228294034279293, −9.927024328009443525797820739960, −8.316848895164770491356344883305, −6.90607445899138641051915510950, −5.90791978655439757138391249423, −5.32332555503613775903217341612, −3.98427442740634116310155600764, −2.83610359431614616422093255194, −1.60851695304091276572345095134,
1.60851695304091276572345095134, 2.83610359431614616422093255194, 3.98427442740634116310155600764, 5.32332555503613775903217341612, 5.90791978655439757138391249423, 6.90607445899138641051915510950, 8.316848895164770491356344883305, 9.927024328009443525797820739960, 10.80859912131595228294034279293, 11.80715044539871697007724302341
