| L(s) = 1 | + 3.30·2-s + 0.708·3-s + 2.92·4-s + 6.96·5-s + 2.34·6-s − 22.1·7-s − 16.7·8-s − 26.4·9-s + 23.0·10-s − 11·11-s + 2.07·12-s − 73.1·14-s + 4.93·15-s − 78.8·16-s + 86.4·17-s − 87.5·18-s + 151.·19-s + 20.3·20-s − 15.6·21-s − 36.3·22-s + 3.92·23-s − 11.8·24-s − 76.4·25-s − 37.8·27-s − 64.6·28-s − 216.·29-s + 16.3·30-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.136·3-s + 0.365·4-s + 0.623·5-s + 0.159·6-s − 1.19·7-s − 0.741·8-s − 0.981·9-s + 0.728·10-s − 0.301·11-s + 0.0498·12-s − 1.39·14-s + 0.0849·15-s − 1.23·16-s + 1.23·17-s − 1.14·18-s + 1.82·19-s + 0.227·20-s − 0.162·21-s − 0.352·22-s + 0.0355·23-s − 0.101·24-s − 0.611·25-s − 0.270·27-s − 0.436·28-s − 1.38·29-s + 0.0992·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.050821416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.050821416\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.30T + 8T^{2} \) |
| 3 | \( 1 - 0.708T + 27T^{2} \) |
| 5 | \( 1 - 6.96T + 125T^{2} \) |
| 7 | \( 1 + 22.1T + 343T^{2} \) |
| 17 | \( 1 - 86.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 3.92T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 21.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 23.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 336.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 473.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 442.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 839.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 807.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 962.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 919.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 257.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 14.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−9.092620829672904233641942240298, −8.023831550037126562993736211417, −7.08677121775494705172985730420, −6.01695744031459734475824962345, −5.70773517579002795314852809200, −5.01464884126715824215966178539, −3.60375733346209966716815757148, −3.23345047427566223381153866475, −2.36133474410920990440970023830, −0.65274502321674712922044159675,
0.65274502321674712922044159675, 2.36133474410920990440970023830, 3.23345047427566223381153866475, 3.60375733346209966716815757148, 5.01464884126715824215966178539, 5.70773517579002795314852809200, 6.01695744031459734475824962345, 7.08677121775494705172985730420, 8.023831550037126562993736211417, 9.092620829672904233641942240298
