| L(s) = 1 | + 18.2·2-s − 27·3-s + 205.·4-s − 55.2·5-s − 492.·6-s − 1.25e3·7-s + 1.40e3·8-s + 729·9-s − 1.00e3·10-s − 4.57e3·11-s − 5.53e3·12-s − 2.12e3·13-s − 2.29e4·14-s + 1.49e3·15-s − 552.·16-s − 4.07e3·17-s + 1.33e4·18-s + 1.32e4·19-s − 1.13e4·20-s + 3.39e4·21-s − 8.34e4·22-s + 1.21e4·23-s − 3.80e4·24-s − 7.50e4·25-s − 3.87e4·26-s − 1.96e4·27-s − 2.58e5·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 0.577·3-s + 1.60·4-s − 0.197·5-s − 0.931·6-s − 1.38·7-s + 0.972·8-s + 0.333·9-s − 0.318·10-s − 1.03·11-s − 0.925·12-s − 0.268·13-s − 2.23·14-s + 0.114·15-s − 0.0336·16-s − 0.200·17-s + 0.537·18-s + 0.444·19-s − 0.316·20-s + 0.800·21-s − 1.67·22-s + 0.208·23-s − 0.561·24-s − 0.960·25-s − 0.432·26-s − 0.192·27-s − 2.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 18.2T + 128T^{2} \) |
| 5 | \( 1 + 55.2T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.25e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.57e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.12e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.07e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.32e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.59e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.47e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.46e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.53e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.46e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.28e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.51e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.74e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.01e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.24e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.45e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.22e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.95e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.62e6T + 8.07e13T^{2} \) |
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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
−12.90705714434074756706331653047, −12.02182226292281744104115973345, −10.87868578128414533571615551900, −9.580206149718944070940376745736, −7.39637002539725286301113543080, −6.22486204712772757264146713695, −5.29004798107548688080606984433, −3.89783279373140133343929921176, −2.66026415421870153907408770417, 0,
2.66026415421870153907408770417, 3.89783279373140133343929921176, 5.29004798107548688080606984433, 6.22486204712772757264146713695, 7.39637002539725286301113543080, 9.580206149718944070940376745736, 10.87868578128414533571615551900, 12.02182226292281744104115973345, 12.90705714434074756706331653047
