| L(s) = 1 | + 1.89·2-s + 2.95·3-s − 4.42·4-s − 1.51·5-s + 5.58·6-s + 5.94·7-s − 23.4·8-s − 18.2·9-s − 2.86·10-s + 11.5·11-s − 13.0·12-s + 22.6·13-s + 11.2·14-s − 4.48·15-s − 9.02·16-s + 120.·17-s − 34.5·18-s − 19·19-s + 6.71·20-s + 17.5·21-s + 21.7·22-s + 63.9·23-s − 69.3·24-s − 122.·25-s + 42.7·26-s − 133.·27-s − 26.2·28-s + ⋯ |
| L(s) = 1 | + 0.668·2-s + 0.568·3-s − 0.553·4-s − 0.135·5-s + 0.380·6-s + 0.320·7-s − 1.03·8-s − 0.676·9-s − 0.0907·10-s + 0.315·11-s − 0.314·12-s + 0.482·13-s + 0.214·14-s − 0.0771·15-s − 0.140·16-s + 1.71·17-s − 0.452·18-s − 0.229·19-s + 0.0750·20-s + 0.182·21-s + 0.211·22-s + 0.579·23-s − 0.590·24-s − 0.981·25-s + 0.322·26-s − 0.953·27-s − 0.177·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.395025057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395025057\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 1.89T + 8T^{2} \) |
| 3 | \( 1 - 2.95T + 27T^{2} \) |
| 5 | \( 1 + 1.51T + 125T^{2} \) |
| 7 | \( 1 - 5.94T + 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 120.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 63.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 373.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 364.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 816.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 220.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 383.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 537.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 616.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 524.T + 9.12e5T^{2} \) |
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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
−18.13746862586972396123275398115, −16.74143017614854947186662895907, −14.89072574467196254879323755163, −14.24330505534209114655038697667, −12.98894972556472582823645446400, −11.53949360615419159706432468369, −9.438289098868931336409521284559, −8.093737082628268959245413428410, −5.60155656187740077362057676202, −3.58036129084453324196323961651,
3.58036129084453324196323961651, 5.60155656187740077362057676202, 8.093737082628268959245413428410, 9.438289098868931336409521284559, 11.53949360615419159706432468369, 12.98894972556472582823645446400, 14.24330505534209114655038697667, 14.89072574467196254879323755163, 16.74143017614854947186662895907, 18.13746862586972396123275398115
