| L(s) = 1 | − 54.5·2-s + 1.61e4·3-s − 1.28e5·4-s + 1.56e5·5-s − 8.78e5·6-s − 4.97e5·7-s + 1.41e7·8-s + 1.30e8·9-s − 8.51e6·10-s + 5.54e8·11-s − 2.06e9·12-s + 2.95e7·13-s + 2.71e7·14-s + 2.51e9·15-s + 1.60e10·16-s + 1.08e10·17-s − 7.10e9·18-s + 7.29e10·19-s − 2.00e10·20-s − 8.01e9·21-s − 3.02e10·22-s − 5.06e11·23-s + 2.27e11·24-s − 7.38e11·25-s − 1.61e9·26-s + 1.72e10·27-s + 6.37e10·28-s + ⋯ |
| L(s) = 1 | − 0.150·2-s + 1.41·3-s − 0.977·4-s + 0.178·5-s − 0.213·6-s − 0.0326·7-s + 0.297·8-s + 1.00·9-s − 0.0269·10-s + 0.780·11-s − 1.38·12-s + 0.0100·13-s + 0.00491·14-s + 0.253·15-s + 0.932·16-s + 0.378·17-s − 0.151·18-s + 0.985·19-s − 0.174·20-s − 0.0462·21-s − 0.117·22-s − 1.34·23-s + 0.422·24-s − 0.968·25-s − 0.00151·26-s + 0.0117·27-s + 0.0318·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(3.048603713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.048603713\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 - 1.46e14T \) |
| good | 2 | \( 1 + 54.5T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.61e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.56e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 4.97e5T + 2.32e14T^{2} \) |
| 11 | \( 1 - 5.54e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.95e7T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.08e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 7.29e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.06e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.68e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 4.20e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.01e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.21e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.10e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.18e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.67e14T + 2.05e29T^{2} \) |
| 61 | \( 1 - 1.32e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.42e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 5.82e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.87e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.61e14T + 1.81e32T^{2} \) |
| 83 | \( 1 + 7.54e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.80e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 3.23e16T + 5.95e33T^{2} \) |
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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
−11.82682614524758873090086473377, −9.926365454711038956326815216326, −9.408052433442789331543078785911, −8.374906520662036219978817997023, −7.57058042173173370873142545910, −5.79175964052071279990514259085, −4.22083462912920923533095826415, −3.43048170082184716261325345793, −2.06856523431170722901599857346, −0.822645335424478244039815889044,
0.822645335424478244039815889044, 2.06856523431170722901599857346, 3.43048170082184716261325345793, 4.22083462912920923533095826415, 5.79175964052071279990514259085, 7.57058042173173370873142545910, 8.374906520662036219978817997023, 9.408052433442789331543078785911, 9.926365454711038956326815216326, 11.82682614524758873090086473377
