| L(s) = 1 | − 3.14e5·2-s − 1.29e8·3-s + 6.47e10·4-s + 2.58e12·5-s + 4.06e13·6-s + 8.89e14·7-s − 9.58e15·8-s + 1.66e16·9-s − 8.14e17·10-s + 2.05e18·11-s − 8.36e18·12-s − 1.82e19·13-s − 2.79e20·14-s − 3.33e20·15-s + 7.91e20·16-s + 3.53e21·17-s − 5.25e21·18-s + 2.01e22·19-s + 1.67e23·20-s − 1.14e23·21-s − 6.46e23·22-s + 2.51e23·23-s + 1.23e24·24-s + 3.77e24·25-s + 5.73e24·26-s − 2.15e24·27-s + 5.76e25·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.88·4-s + 1.51·5-s + 0.980·6-s + 1.44·7-s − 1.50·8-s + 0.333·9-s − 2.57·10-s + 1.22·11-s − 1.08·12-s − 0.583·13-s − 2.45·14-s − 0.875·15-s + 0.670·16-s + 1.03·17-s − 0.566·18-s + 0.844·19-s + 2.85·20-s − 0.833·21-s − 2.07·22-s + 0.371·23-s + 0.868·24-s + 1.29·25-s + 0.991·26-s − 0.192·27-s + 2.72·28-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
| \(L(18)\) |
\(\approx\) |
\(1.247530533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247530533\) |
| \(L(\frac{37}{2})\) |
|
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 | \( 1 + 3.14e5T + 3.43e10T^{2} \) |
| 5 | \( 1 - 2.58e12T + 2.91e24T^{2} \) |
| 7 | \( 1 - 8.89e14T + 3.78e29T^{2} \) |
| 11 | \( 1 - 2.05e18T + 2.81e36T^{2} \) |
| 13 | \( 1 + 1.82e19T + 9.72e38T^{2} \) |
| 17 | \( 1 - 3.53e21T + 1.16e43T^{2} \) |
| 19 | \( 1 - 2.01e22T + 5.70e44T^{2} \) |
| 23 | \( 1 - 2.51e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 8.93e24T + 1.52e51T^{2} \) |
| 31 | \( 1 + 1.94e26T + 1.57e52T^{2} \) |
| 37 | \( 1 + 2.17e27T + 7.71e54T^{2} \) |
| 41 | \( 1 + 7.70e27T + 2.80e56T^{2} \) |
| 43 | \( 1 + 2.60e28T + 1.48e57T^{2} \) |
| 47 | \( 1 - 3.33e29T + 3.33e58T^{2} \) |
| 53 | \( 1 - 1.01e30T + 2.23e60T^{2} \) |
| 59 | \( 1 - 1.91e30T + 9.54e61T^{2} \) |
| 61 | \( 1 - 7.04e30T + 3.06e62T^{2} \) |
| 67 | \( 1 - 1.18e32T + 8.17e63T^{2} \) |
| 71 | \( 1 + 2.78e32T + 6.22e64T^{2} \) |
| 73 | \( 1 + 3.82e32T + 1.64e65T^{2} \) |
| 79 | \( 1 + 1.32e33T + 2.61e66T^{2} \) |
| 83 | \( 1 + 5.69e33T + 1.47e67T^{2} \) |
| 89 | \( 1 - 1.72e34T + 1.69e68T^{2} \) |
| 97 | \( 1 - 4.83e34T + 3.44e69T^{2} \) |
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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
−17.46411308228721660928474700715, −16.87243264750390649437292282361, −14.37947788642795473564157206699, −11.65320764150379002964282650479, −10.23224323759285735384010866733, −9.048631541195668366788541643019, −7.22355413379947825433517940696, −5.47597542128395986538455733094, −1.85550417948922011767130155612, −1.11892760295465658416861557080,
1.11892760295465658416861557080, 1.85550417948922011767130155612, 5.47597542128395986538455733094, 7.22355413379947825433517940696, 9.048631541195668366788541643019, 10.23224323759285735384010866733, 11.65320764150379002964282650479, 14.37947788642795473564157206699, 16.87243264750390649437292282361, 17.46411308228721660928474700715
