| L(s) = 1 | − 4.16e4·2-s − 4.78e6·3-s + 1.19e9·4-s + 9.35e9·5-s + 1.99e11·6-s + 6.78e11·7-s − 2.74e13·8-s + 2.28e13·9-s − 3.89e14·10-s − 5.18e14·11-s − 5.72e15·12-s − 9.78e15·13-s − 2.82e16·14-s − 4.47e16·15-s + 5.01e17·16-s − 9.17e17·17-s − 9.52e17·18-s − 5.21e17·19-s + 1.11e19·20-s − 3.24e18·21-s + 2.15e19·22-s + 1.03e20·23-s + 1.31e20·24-s − 9.87e19·25-s + 4.07e20·26-s − 1.09e20·27-s + 8.11e20·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 0.577·3-s + 2.22·4-s + 0.685·5-s + 1.03·6-s + 0.377·7-s − 2.20·8-s + 0.333·9-s − 1.23·10-s − 0.411·11-s − 1.28·12-s − 0.689·13-s − 0.679·14-s − 0.395·15-s + 1.73·16-s − 1.32·17-s − 0.598·18-s − 0.149·19-s + 1.52·20-s − 0.218·21-s + 0.740·22-s + 1.85·23-s + 1.27·24-s − 0.530·25-s + 1.23·26-s − 0.192·27-s + 0.842·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.6007969968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6007969968\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4.78e6T \) |
| 7 | \( 1 - 6.78e11T \) |
| good | 2 | \( 1 + 4.16e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 9.35e9T + 1.86e20T^{2} \) |
| 11 | \( 1 + 5.18e14T + 1.58e30T^{2} \) |
| 13 | \( 1 + 9.78e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + 9.17e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 5.21e17T + 1.21e37T^{2} \) |
| 23 | \( 1 - 1.03e20T + 3.09e39T^{2} \) |
| 29 | \( 1 - 1.65e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + 4.31e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 5.32e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 8.95e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 3.51e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 9.32e22T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.50e25T + 1.00e50T^{2} \) |
| 59 | \( 1 - 6.90e25T + 2.26e51T^{2} \) |
| 61 | \( 1 - 4.68e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 9.86e25T + 9.04e52T^{2} \) |
| 71 | \( 1 + 7.36e26T + 4.85e53T^{2} \) |
| 73 | \( 1 - 1.98e26T + 1.08e54T^{2} \) |
| 79 | \( 1 + 9.75e26T + 1.07e55T^{2} \) |
| 83 | \( 1 + 1.15e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 2.78e28T + 3.40e56T^{2} \) |
| 97 | \( 1 - 3.03e28T + 4.13e57T^{2} \) |
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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
−11.51575047999271400426558842904, −10.68376652166833406837623617135, −9.680422810283334290463950251136, −8.678663507951299514664553156871, −7.36353739658666882745336391271, −6.40459114216201165872637196671, −4.95224229320079142226896984523, −2.57797784343392557444159049568, −1.63567414241960552305670038608, −0.49803418075559936191300793391,
0.49803418075559936191300793391, 1.63567414241960552305670038608, 2.57797784343392557444159049568, 4.95224229320079142226896984523, 6.40459114216201165872637196671, 7.36353739658666882745336391271, 8.678663507951299514664553156871, 9.680422810283334290463950251136, 10.68376652166833406837623617135, 11.51575047999271400426558842904
