| L(s) = 1 | − 8.42·2-s + 9·3-s + 38.9·4-s + 57.0·5-s − 75.7·6-s + 139.·7-s − 58.3·8-s + 81·9-s − 480.·10-s + 317.·11-s + 350.·12-s − 613.·13-s − 1.17e3·14-s + 513.·15-s − 754.·16-s − 240.·17-s − 682.·18-s − 199.·19-s + 2.22e3·20-s + 1.25e3·21-s − 2.67e3·22-s + 3.93e3·23-s − 525.·24-s + 130.·25-s + 5.16e3·26-s + 729·27-s + 5.44e3·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.21·4-s + 1.02·5-s − 0.859·6-s + 1.07·7-s − 0.322·8-s + 0.333·9-s − 1.51·10-s + 0.792·11-s + 0.702·12-s − 1.00·13-s − 1.60·14-s + 0.589·15-s − 0.736·16-s − 0.201·17-s − 0.496·18-s − 0.126·19-s + 1.24·20-s + 0.623·21-s − 1.17·22-s + 1.55·23-s − 0.186·24-s + 0.0416·25-s + 1.49·26-s + 0.192·27-s + 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.446384926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.446384926\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 29 | \( 1 + 841T \) |
| good | 2 | \( 1 + 8.42T + 32T^{2} \) |
| 5 | \( 1 - 57.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 139.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 317.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 613.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 240.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 199.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.93e3T + 6.43e6T^{2} \) |
| 31 | \( 1 - 8.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.59e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.64e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.66e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.11e5T + 8.58e9T^{2} \) |
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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
−13.41672048567187819102552476486, −11.80098147923313845515279461918, −10.62516117102216497707786978872, −9.640035565688429866631894265511, −8.911808765211502410540227565676, −7.86484499367927754059477526875, −6.71757990585547941010166695158, −4.81287863514408280621142995424, −2.32798279090917205335546572273, −1.21863244724271810270813613623,
1.21863244724271810270813613623, 2.32798279090917205335546572273, 4.81287863514408280621142995424, 6.71757990585547941010166695158, 7.86484499367927754059477526875, 8.911808765211502410540227565676, 9.640035565688429866631894265511, 10.62516117102216497707786978872, 11.80098147923313845515279461918, 13.41672048567187819102552476486
