| L(s) = 1 | + 1.47·2-s − 5.81·4-s + 11.3·5-s − 28.3·7-s − 20.4·8-s + 16.7·10-s − 56.2·11-s + 24.3·13-s − 41.9·14-s + 16.4·16-s + 17·17-s − 34.0·19-s − 65.9·20-s − 83.0·22-s − 108.·23-s + 3.55·25-s + 36.0·26-s + 165.·28-s − 266.·29-s − 207.·31-s + 187.·32-s + 25.1·34-s − 321.·35-s + 380.·37-s − 50.2·38-s − 231.·40-s + 451.·41-s + ⋯ |
| L(s) = 1 | + 0.522·2-s − 0.727·4-s + 1.01·5-s − 1.53·7-s − 0.901·8-s + 0.529·10-s − 1.54·11-s + 0.520·13-s − 0.800·14-s + 0.256·16-s + 0.242·17-s − 0.410·19-s − 0.737·20-s − 0.804·22-s − 0.982·23-s + 0.0284·25-s + 0.271·26-s + 1.11·28-s − 1.70·29-s − 1.20·31-s + 1.03·32-s + 0.126·34-s − 1.55·35-s + 1.68·37-s − 0.214·38-s − 0.914·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 1.47T + 8T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 + 28.3T + 343T^{2} \) |
| 11 | \( 1 + 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 34.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 266.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 380.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 179.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 56.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 265.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 704.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 509.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 710.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 834.T + 9.12e5T^{2} \) |
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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
−12.70150392833923993168985048737, −10.84234923602669132307129689891, −9.742029771866160122736792023100, −9.284618405104493347295221023592, −7.77153077594818912779379293826, −6.04687634561477708950953865143, −5.62869598682938290675957681616, −3.93722421679993483925939503786, −2.61568117400199109494512535237, 0,
2.61568117400199109494512535237, 3.93722421679993483925939503786, 5.62869598682938290675957681616, 6.04687634561477708950953865143, 7.77153077594818912779379293826, 9.284618405104493347295221023592, 9.742029771866160122736792023100, 10.84234923602669132307129689891, 12.70150392833923993168985048737
