| L(s) = 1 | + 1.86e5·2-s − 1.29e8·3-s + 3.53e8·4-s − 2.21e12·5-s − 2.40e13·6-s + 5.41e14·7-s − 6.33e15·8-s + 1.66e16·9-s − 4.12e17·10-s + 3.21e18·11-s − 4.56e16·12-s + 2.69e19·13-s + 1.00e20·14-s + 2.85e20·15-s − 1.19e21·16-s + 7.92e20·17-s + 3.10e21·18-s − 1.88e22·19-s − 7.82e20·20-s − 6.99e22·21-s + 5.99e23·22-s + 4.62e23·23-s + 8.18e23·24-s + 1.98e24·25-s + 5.02e24·26-s − 2.15e24·27-s + 1.91e23·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.577·3-s + 0.0102·4-s − 1.29·5-s − 0.580·6-s + 0.880·7-s − 0.994·8-s + 0.333·9-s − 1.30·10-s + 1.91·11-s − 0.00594·12-s + 0.864·13-s + 0.884·14-s + 0.749·15-s − 1.01·16-s + 0.232·17-s + 0.335·18-s − 0.787·19-s − 0.0133·20-s − 0.508·21-s + 1.92·22-s + 0.683·23-s + 0.574·24-s + 0.683·25-s + 0.868·26-s − 0.192·27-s + 0.00906·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(18)\) |
\(\approx\) |
\(2.193722188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193722188\) |
| \(L(\frac{37}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.29e8T \) |
| good | 2 | \( 1 - 1.86e5T + 3.43e10T^{2} \) |
| 5 | \( 1 + 2.21e12T + 2.91e24T^{2} \) |
| 7 | \( 1 - 5.41e14T + 3.78e29T^{2} \) |
| 11 | \( 1 - 3.21e18T + 2.81e36T^{2} \) |
| 13 | \( 1 - 2.69e19T + 9.72e38T^{2} \) |
| 17 | \( 1 - 7.92e20T + 1.16e43T^{2} \) |
| 19 | \( 1 + 1.88e22T + 5.70e44T^{2} \) |
| 23 | \( 1 - 4.62e23T + 4.57e47T^{2} \) |
| 29 | \( 1 - 3.40e25T + 1.52e51T^{2} \) |
| 31 | \( 1 - 2.21e26T + 1.57e52T^{2} \) |
| 37 | \( 1 + 2.96e27T + 7.71e54T^{2} \) |
| 41 | \( 1 + 6.83e27T + 2.80e56T^{2} \) |
| 43 | \( 1 - 5.22e28T + 1.48e57T^{2} \) |
| 47 | \( 1 - 3.56e28T + 3.33e58T^{2} \) |
| 53 | \( 1 + 5.00e29T + 2.23e60T^{2} \) |
| 59 | \( 1 - 1.24e31T + 9.54e61T^{2} \) |
| 61 | \( 1 + 2.00e31T + 3.06e62T^{2} \) |
| 67 | \( 1 - 1.01e32T + 8.17e63T^{2} \) |
| 71 | \( 1 - 3.89e32T + 6.22e64T^{2} \) |
| 73 | \( 1 + 2.91e32T + 1.64e65T^{2} \) |
| 79 | \( 1 + 1.06e33T + 2.61e66T^{2} \) |
| 83 | \( 1 - 6.15e33T + 1.47e67T^{2} \) |
| 89 | \( 1 + 1.61e34T + 1.69e68T^{2} \) |
| 97 | \( 1 - 5.66e34T + 3.44e69T^{2} \) |
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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
−17.36686377091751923481858743361, −15.47095812985426385634500212579, −14.19240448294169875785518283962, −12.19870778338905119894522172707, −11.36718097595476200509409895634, −8.574298396327967864015148853907, −6.47502553658849680481827090440, −4.61535285828647644633611824198, −3.73661901778896157798510472506, −0.934202417721700838595969776949,
0.934202417721700838595969776949, 3.73661901778896157798510472506, 4.61535285828647644633611824198, 6.47502553658849680481827090440, 8.574298396327967864015148853907, 11.36718097595476200509409895634, 12.19870778338905119894522172707, 14.19240448294169875785518283962, 15.47095812985426385634500212579, 17.36686377091751923481858743361
