| L(s) = 1 | − 5.24e5·2-s + 2.11e9·3-s + 2.74e11·4-s + 6.99e13·5-s − 1.10e15·6-s + 5.58e16·7-s − 1.44e17·8-s + 4.06e17·9-s − 3.66e19·10-s − 2.13e20·11-s + 5.80e20·12-s + 1.59e21·13-s − 2.92e22·14-s + 1.47e23·15-s + 7.55e22·16-s − 2.01e22·17-s − 2.13e23·18-s + 7.01e23·19-s + 1.92e25·20-s + 1.17e26·21-s + 1.11e26·22-s + 1.72e26·23-s − 3.04e26·24-s + 3.07e27·25-s − 8.34e26·26-s − 7.69e27·27-s + 1.53e28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s + 1.64·5-s − 0.741·6-s + 1.85·7-s − 0.353·8-s + 0.100·9-s − 1.15·10-s − 1.05·11-s + 0.524·12-s + 0.301·13-s − 1.31·14-s + 1.72·15-s + 0.250·16-s − 0.0204·17-s − 0.0709·18-s + 0.0813·19-s + 0.820·20-s + 1.94·21-s + 0.743·22-s + 0.483·23-s − 0.370·24-s + 1.69·25-s − 0.213·26-s − 0.943·27-s + 0.926·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(\approx\) |
\(2.882465862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.882465862\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.24e5T \) |
| good | 3 | \( 1 - 2.11e9T + 4.05e18T^{2} \) |
| 5 | \( 1 - 6.99e13T + 1.81e27T^{2} \) |
| 7 | \( 1 - 5.58e16T + 9.09e32T^{2} \) |
| 11 | \( 1 + 2.13e20T + 4.11e40T^{2} \) |
| 13 | \( 1 - 1.59e21T + 2.77e43T^{2} \) |
| 17 | \( 1 + 2.01e22T + 9.71e47T^{2} \) |
| 19 | \( 1 - 7.01e23T + 7.43e49T^{2} \) |
| 23 | \( 1 - 1.72e26T + 1.28e53T^{2} \) |
| 29 | \( 1 + 3.91e28T + 1.08e57T^{2} \) |
| 31 | \( 1 - 1.95e28T + 1.45e58T^{2} \) |
| 37 | \( 1 + 2.47e30T + 1.44e61T^{2} \) |
| 41 | \( 1 - 3.59e31T + 7.91e62T^{2} \) |
| 43 | \( 1 + 4.72e31T + 5.07e63T^{2} \) |
| 47 | \( 1 - 2.11e32T + 1.62e65T^{2} \) |
| 53 | \( 1 + 2.04e33T + 1.76e67T^{2} \) |
| 59 | \( 1 + 8.82e33T + 1.15e69T^{2} \) |
| 61 | \( 1 + 1.48e33T + 4.24e69T^{2} \) |
| 67 | \( 1 + 5.72e35T + 1.64e71T^{2} \) |
| 71 | \( 1 + 7.39e35T + 1.58e72T^{2} \) |
| 73 | \( 1 - 3.13e36T + 4.67e72T^{2} \) |
| 79 | \( 1 - 1.73e37T + 1.01e74T^{2} \) |
| 83 | \( 1 + 2.82e37T + 6.98e74T^{2} \) |
| 89 | \( 1 - 1.03e38T + 1.06e76T^{2} \) |
| 97 | \( 1 - 3.18e38T + 3.04e77T^{2} \) |
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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
−18.36942423794841135476684376943, −17.34455450695660601251387542759, −14.80190207193498400660355503034, −13.61714537646484465701077158765, −10.78877334774693993085290941082, −9.120765272067143291108306653761, −7.87724456721961639461677820872, −5.40983787226348957427096627134, −2.46169254640360669407218558479, −1.55619428612733078561671947902,
1.55619428612733078561671947902, 2.46169254640360669407218558479, 5.40983787226348957427096627134, 7.87724456721961639461677820872, 9.120765272067143291108306653761, 10.78877334774693993085290941082, 13.61714537646484465701077158765, 14.80190207193498400660355503034, 17.34455450695660601251387542759, 18.36942423794841135476684376943
