| L(s) = 1 | + 2·2-s + 7.87·3-s + 4·4-s + 8.82·5-s + 15.7·6-s − 33.6·7-s + 8·8-s + 35.0·9-s + 17.6·10-s − 66.5·11-s + 31.5·12-s + 4.52·13-s − 67.3·14-s + 69.5·15-s + 16·16-s + 105.·17-s + 70.0·18-s − 53.3·19-s + 35.3·20-s − 265.·21-s − 133.·22-s + 161.·23-s + 63.0·24-s − 47.0·25-s + 9.05·26-s + 63.2·27-s − 134.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.789·5-s + 1.07·6-s − 1.81·7-s + 0.353·8-s + 1.29·9-s + 0.558·10-s − 1.82·11-s + 0.757·12-s + 0.0966·13-s − 1.28·14-s + 1.19·15-s + 0.250·16-s + 1.50·17-s + 0.917·18-s − 0.643·19-s + 0.394·20-s − 2.75·21-s − 1.28·22-s + 1.46·23-s + 0.535·24-s − 0.376·25-s + 0.0683·26-s + 0.451·27-s − 0.908·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.075076802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.075076802\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 37 | \( 1 - 37T \) |
| good | 3 | \( 1 - 7.87T + 27T^{2} \) |
| 5 | \( 1 - 8.82T + 125T^{2} \) |
| 7 | \( 1 + 33.6T + 343T^{2} \) |
| 11 | \( 1 + 66.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.52T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 28.9T + 2.97e4T^{2} \) |
| 41 | \( 1 + 446.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 174.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 130.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 275.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 627.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 23.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 407.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 49.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 832.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 816.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 28.0T + 9.12e5T^{2} \) |
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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
−13.78400830179480479323842805469, −13.23565825656344914806812244463, −12.60109769179049113292401973380, −10.29405099157376687273412147423, −9.708020540113319345878004714835, −8.297439019430225493316775507062, −6.94597415030121571555139414599, −5.50343274454870716552099385767, −3.35327857471839286844546667230, −2.59790012798435350382181632576,
2.59790012798435350382181632576, 3.35327857471839286844546667230, 5.50343274454870716552099385767, 6.94597415030121571555139414599, 8.297439019430225493316775507062, 9.708020540113319345878004714835, 10.29405099157376687273412147423, 12.60109769179049113292401973380, 13.23565825656344914806812244463, 13.78400830179480479323842805469
