| L(s) = 1 | − 27·3-s − 170.·5-s − 803.·7-s + 729·9-s + 8.55e3·11-s + 1.46e3·13-s + 4.59e3·15-s + 3.07e4·17-s + 1.13e4·19-s + 2.16e4·21-s − 1.16e5·23-s − 4.91e4·25-s − 1.96e4·27-s − 1.84e5·29-s + 6.13e4·31-s − 2.31e5·33-s + 1.36e5·35-s − 3.57e5·37-s − 3.95e4·39-s + 3.89e5·41-s − 9.39e5·43-s − 1.24e5·45-s + 1.13e6·47-s − 1.78e5·49-s − 8.31e5·51-s − 3.24e5·53-s − 1.45e6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.609·5-s − 0.885·7-s + 0.333·9-s + 1.93·11-s + 0.185·13-s + 0.351·15-s + 1.51·17-s + 0.381·19-s + 0.511·21-s − 1.99·23-s − 0.628·25-s − 0.192·27-s − 1.40·29-s + 0.369·31-s − 1.11·33-s + 0.539·35-s − 1.15·37-s − 0.106·39-s + 0.882·41-s − 1.80·43-s − 0.203·45-s + 1.60·47-s − 0.216·49-s − 0.877·51-s − 0.299·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.287208379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287208379\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| good | 5 | \( 1 + 170.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 803.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.55e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.46e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.13e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.16e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.84e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.13e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.57e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.24e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.86e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.58e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.98e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.61e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.61e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.956088978111212042715041156868, −9.517943590020345913965390358757, −8.245266831046170513562640344614, −7.22269889516030153189954758515, −6.32868809843472846236634185654, −5.55724193137623994777848992418, −3.91909952348883134840812049173, −3.60950761769486532779291438849, −1.69406103942966313962299964683, −0.55813324305753457303473788349,
0.55813324305753457303473788349, 1.69406103942966313962299964683, 3.60950761769486532779291438849, 3.91909952348883134840812049173, 5.55724193137623994777848992418, 6.32868809843472846236634185654, 7.22269889516030153189954758515, 8.245266831046170513562640344614, 9.517943590020345913965390358757, 9.956088978111212042715041156868
