L(s) = 1 | − 10.7·2-s − 12.1·4-s − 451.·5-s − 504.·7-s + 1.50e3·8-s + 4.86e3·10-s + 8.07e3·11-s − 4.67e3·13-s + 5.43e3·14-s − 1.46e4·16-s + 1.34e4·17-s − 1.13e4·19-s + 5.49e3·20-s − 8.68e4·22-s + 6.64e4·23-s + 1.26e5·25-s + 5.03e4·26-s + 6.13e3·28-s + 1.74e5·29-s + 6.24e4·31-s − 3.50e4·32-s − 1.45e5·34-s + 2.28e5·35-s + 2.77e5·37-s + 1.22e5·38-s − 6.81e5·40-s − 5.29e5·41-s + ⋯ |
L(s) = 1 | − 0.951·2-s − 0.0949·4-s − 1.61·5-s − 0.556·7-s + 1.04·8-s + 1.53·10-s + 1.82·11-s − 0.590·13-s + 0.529·14-s − 0.895·16-s + 0.666·17-s − 0.380·19-s + 0.153·20-s − 1.73·22-s + 1.13·23-s + 1.61·25-s + 0.561·26-s + 0.0528·28-s + 1.32·29-s + 0.376·31-s − 0.189·32-s − 0.633·34-s + 0.899·35-s + 0.900·37-s + 0.361·38-s − 1.68·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.7552545269\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7552545269\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 + 10.7T + 128T^{2} \) |
| 5 | \( 1 + 451.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 504.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.34e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.13e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.74e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.24e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.75e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.13e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 3.01e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.85e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.34e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.94e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.72e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.67e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.89e6T + 8.07e13T^{2} \) |
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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
−9.493823320795331860124943691633, −8.843176294174679851966829658771, −8.047097111083515940336865168548, −7.22883598150130881643508087291, −6.49411369668523316349577189127, −4.73240808886345491389832516737, −4.05593742147713795927120572011, −3.08947043013686246643827966540, −1.25394845255747029777790641318, −0.51703408605922632415201744884,
0.51703408605922632415201744884, 1.25394845255747029777790641318, 3.08947043013686246643827966540, 4.05593742147713795927120572011, 4.73240808886345491389832516737, 6.49411369668523316349577189127, 7.22883598150130881643508087291, 8.047097111083515940336865168548, 8.843176294174679851966829658771, 9.493823320795331860124943691633