| L(s) = 1 | − 6.40·2-s − 9·3-s + 9.06·4-s − 77.9·5-s + 57.6·6-s + 112.·7-s + 146.·8-s + 81·9-s + 499.·10-s − 791.·11-s − 81.5·12-s − 741.·13-s − 719.·14-s + 701.·15-s − 1.23e3·16-s − 1.11e3·17-s − 519.·18-s − 1.17e3·19-s − 706.·20-s − 1.01e3·21-s + 5.07e3·22-s − 3.48e3·23-s − 1.32e3·24-s + 2.94e3·25-s + 4.74e3·26-s − 729·27-s + 1.01e3·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.283·4-s − 1.39·5-s + 0.654·6-s + 0.866·7-s + 0.811·8-s + 0.333·9-s + 1.57·10-s − 1.97·11-s − 0.163·12-s − 1.21·13-s − 0.981·14-s + 0.804·15-s − 1.20·16-s − 0.939·17-s − 0.377·18-s − 0.748·19-s − 0.394·20-s − 0.500·21-s + 2.23·22-s − 1.37·23-s − 0.468·24-s + 0.942·25-s + 1.37·26-s − 0.192·27-s + 0.245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03712846926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03712846926\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 6.40T + 32T^{2} \) |
| 5 | \( 1 + 77.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 112.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 791.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 741.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.11e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 9.70e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.37e5T + 8.58e9T^{2} \) |
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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
−11.50187057994022656079855506791, −10.71660239635891997558721644310, −9.988780802391077990704170470718, −8.379955928508461750914370140284, −7.940349396914995393980982024201, −7.11473455796705599134287223542, −5.10857426761324746938706133666, −4.32953785867813269875330352308, −2.15578377093210361030676336905, −0.14071825729615920649905622193,
0.14071825729615920649905622193, 2.15578377093210361030676336905, 4.32953785867813269875330352308, 5.10857426761324746938706133666, 7.11473455796705599134287223542, 7.940349396914995393980982024201, 8.379955928508461750914370140284, 9.988780802391077990704170470718, 10.71660239635891997558721644310, 11.50187057994022656079855506791
