| L(s) = 1 | − 18.8·2-s + 227.·4-s − 70.1·5-s + 1.06e3·7-s − 1.86e3·8-s + 1.32e3·10-s + 1.45e3·11-s + 9.74e3·13-s − 2.00e4·14-s + 6.11e3·16-s − 2.13e4·17-s + 4.52e4·19-s − 1.59e4·20-s − 2.74e4·22-s + 3.97e4·23-s − 7.32e4·25-s − 1.83e5·26-s + 2.42e5·28-s − 1.06e5·29-s − 1.95e5·31-s + 1.23e5·32-s + 4.02e5·34-s − 7.47e4·35-s − 3.47e4·37-s − 8.52e5·38-s + 1.30e5·40-s − 2.17e5·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.77·4-s − 0.250·5-s + 1.17·7-s − 1.28·8-s + 0.417·10-s + 0.329·11-s + 1.22·13-s − 1.95·14-s + 0.373·16-s − 1.05·17-s + 1.51·19-s − 0.445·20-s − 0.549·22-s + 0.681·23-s − 0.937·25-s − 2.04·26-s + 2.08·28-s − 0.813·29-s − 1.17·31-s + 0.667·32-s + 1.75·34-s − 0.294·35-s − 0.112·37-s − 2.51·38-s + 0.323·40-s − 0.493·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 + 18.8T + 128T^{2} \) |
| 5 | \( 1 + 70.1T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.06e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.74e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.13e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.52e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.95e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.47e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.17e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 9.14e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.04e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.62e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.05e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.51e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.61e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.11e7T + 8.07e13T^{2} \) |
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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
−9.402969439569406203929374253804, −8.843364830506323558152401473127, −7.950895664712532864438245467609, −7.34721719027692048695777671860, −6.21650500489747491791492992218, −4.88024556363347474151881643329, −3.47602391895755534774914479884, −1.85597619390279506440798998679, −1.25646404574567169880882976236, 0,
1.25646404574567169880882976236, 1.85597619390279506440798998679, 3.47602391895755534774914479884, 4.88024556363347474151881643329, 6.21650500489747491791492992218, 7.34721719027692048695777671860, 7.950895664712532864438245467609, 8.843364830506323558152401473127, 9.402969439569406203929374253804
