| L(s) = 1 | − 9.83e5·2-s − 1.16e9·3-s + 4.17e11·4-s + 1.57e13·5-s + 1.14e15·6-s + 5.39e15·7-s + 1.30e17·8-s + 1.35e18·9-s − 1.55e19·10-s − 1.89e20·11-s − 4.85e20·12-s − 1.17e21·13-s − 5.30e21·14-s − 1.83e22·15-s − 3.57e23·16-s − 1.39e24·17-s − 1.32e24·18-s − 2.79e24·19-s + 6.59e24·20-s − 6.26e24·21-s + 1.86e26·22-s + 5.34e26·23-s − 1.51e26·24-s − 1.56e27·25-s + 1.15e27·26-s − 1.57e27·27-s + 2.25e27·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.577·3-s + 0.759·4-s + 0.370·5-s + 0.765·6-s + 0.178·7-s + 0.319·8-s + 0.333·9-s − 0.491·10-s − 0.933·11-s − 0.438·12-s − 0.222·13-s − 0.237·14-s − 0.213·15-s − 1.18·16-s − 1.41·17-s − 0.442·18-s − 0.323·19-s + 0.281·20-s − 0.103·21-s + 1.23·22-s + 1.49·23-s − 0.184·24-s − 0.862·25-s + 0.294·26-s − 0.192·27-s + 0.135·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(\approx\) |
\(0.5853313267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5853313267\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.16e9T \) |
| good | 2 | \( 1 + 9.83e5T + 5.49e11T^{2} \) |
| 5 | \( 1 - 1.57e13T + 1.81e27T^{2} \) |
| 7 | \( 1 - 5.39e15T + 9.09e32T^{2} \) |
| 11 | \( 1 + 1.89e20T + 4.11e40T^{2} \) |
| 13 | \( 1 + 1.17e21T + 2.77e43T^{2} \) |
| 17 | \( 1 + 1.39e24T + 9.71e47T^{2} \) |
| 19 | \( 1 + 2.79e24T + 7.43e49T^{2} \) |
| 23 | \( 1 - 5.34e26T + 1.28e53T^{2} \) |
| 29 | \( 1 - 2.12e28T + 1.08e57T^{2} \) |
| 31 | \( 1 - 1.34e29T + 1.45e58T^{2} \) |
| 37 | \( 1 + 4.34e30T + 1.44e61T^{2} \) |
| 41 | \( 1 - 3.17e31T + 7.91e62T^{2} \) |
| 43 | \( 1 + 8.86e31T + 5.07e63T^{2} \) |
| 47 | \( 1 + 3.38e32T + 1.62e65T^{2} \) |
| 53 | \( 1 - 7.04e33T + 1.76e67T^{2} \) |
| 59 | \( 1 - 1.96e34T + 1.15e69T^{2} \) |
| 61 | \( 1 - 6.79e34T + 4.24e69T^{2} \) |
| 67 | \( 1 - 4.10e35T + 1.64e71T^{2} \) |
| 71 | \( 1 - 8.69e35T + 1.58e72T^{2} \) |
| 73 | \( 1 - 3.39e36T + 4.67e72T^{2} \) |
| 79 | \( 1 + 1.47e37T + 1.01e74T^{2} \) |
| 83 | \( 1 - 1.98e37T + 6.98e74T^{2} \) |
| 89 | \( 1 + 1.95e37T + 1.06e76T^{2} \) |
| 97 | \( 1 - 3.46e38T + 3.04e77T^{2} \) |
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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
−17.21091750328857127986392209997, −15.68446044325042690276775077987, −13.27865707835231704204784554121, −11.15666885629649966618385242900, −9.980572378766690460606941198006, −8.451738440920814554881711828451, −6.84169261746180358147463070705, −4.87852124076909544237890976039, −2.15129261232880695177346048504, −0.59979934643266272629309286477,
0.59979934643266272629309286477, 2.15129261232880695177346048504, 4.87852124076909544237890976039, 6.84169261746180358147463070705, 8.451738440920814554881711828451, 9.980572378766690460606941198006, 11.15666885629649966618385242900, 13.27865707835231704204784554121, 15.68446044325042690276775077987, 17.21091750328857127986392209997
