| L(s) = 1 | − 21.3·2-s + 27·3-s + 325.·4-s + 362.·5-s − 575.·6-s + 343·7-s − 4.21e3·8-s + 729·9-s − 7.71e3·10-s + 5.51e3·11-s + 8.80e3·12-s + 8.63e3·13-s − 7.30e3·14-s + 9.77e3·15-s + 4.81e4·16-s + 1.43e4·17-s − 1.55e4·18-s + 1.07e4·19-s + 1.18e5·20-s + 9.26e3·21-s − 1.17e5·22-s − 1.21e4·23-s − 1.13e5·24-s + 5.30e4·25-s − 1.84e5·26-s + 1.96e4·27-s + 1.11e5·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.577·3-s + 2.54·4-s + 1.29·5-s − 1.08·6-s + 0.377·7-s − 2.91·8-s + 0.333·9-s − 2.44·10-s + 1.24·11-s + 1.47·12-s + 1.09·13-s − 0.711·14-s + 0.748·15-s + 2.93·16-s + 0.708·17-s − 0.627·18-s + 0.358·19-s + 3.30·20-s + 0.218·21-s − 2.35·22-s − 0.208·23-s − 1.68·24-s + 0.678·25-s − 2.05·26-s + 0.192·27-s + 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.258362664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258362664\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 21.3T + 128T^{2} \) |
| 5 | \( 1 - 362.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 5.51e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.63e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.43e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.07e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.19e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.07e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.37e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.16e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.10e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.39e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.76e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.29e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.82e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.97e6T + 8.07e13T^{2} \) |
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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
−9.637436599413032396994447952383, −9.023918030035080297601579499321, −8.370359982132579762881833050713, −7.37600315473592575900014899853, −6.44723993397779418593536472022, −5.69330935707454275082562553313, −3.66817390050551471725242608518, −2.38315507392720889465081564119, −1.52599562923554458877974910374, −0.972215961620210351312620420793,
0.972215961620210351312620420793, 1.52599562923554458877974910374, 2.38315507392720889465081564119, 3.66817390050551471725242608518, 5.69330935707454275082562553313, 6.44723993397779418593536472022, 7.37600315473592575900014899853, 8.370359982132579762881833050713, 9.023918030035080297601579499321, 9.637436599413032396994447952383
