| L(s) = 1 | − 16.7·2-s − 46.6·3-s + 152.·4-s − 389.·5-s + 782.·6-s − 1.15e3·7-s − 413.·8-s − 7.99·9-s + 6.53e3·10-s − 7.12e3·12-s − 2.98e3·13-s + 1.93e4·14-s + 1.81e4·15-s − 1.26e4·16-s − 1.53e4·17-s + 133.·18-s + 1.04e4·19-s − 5.95e4·20-s + 5.38e4·21-s + 1.13e5·23-s + 1.92e4·24-s + 7.38e4·25-s + 4.99e4·26-s + 1.02e5·27-s − 1.76e5·28-s − 3.88e4·29-s − 3.04e5·30-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.998·3-s + 1.19·4-s − 1.39·5-s + 1.47·6-s − 1.27·7-s − 0.285·8-s − 0.00365·9-s + 2.06·10-s − 1.19·12-s − 0.376·13-s + 1.88·14-s + 1.39·15-s − 0.770·16-s − 0.757·17-s + 0.00541·18-s + 0.349·19-s − 1.66·20-s + 1.26·21-s + 1.93·23-s + 0.284·24-s + 0.945·25-s + 0.557·26-s + 1.00·27-s − 1.51·28-s − 0.296·29-s − 2.06·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 16.7T + 128T^{2} \) |
| 3 | \( 1 + 46.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 389.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.15e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 2.98e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.53e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.04e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.13e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.88e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.83e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.69e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.77e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.10e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.58e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.29e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.18e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.54e6T + 8.07e13T^{2} \) |
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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
−11.34880910121064609283660591817, −10.56347601143106942416278207090, −9.432705067809431215474966666216, −8.447693182029751111158905221001, −7.24704027093279717365420605400, −6.50213031735933206148033533821, −4.72814705073561007578831702072, −3.04353762371028671316761287882, −0.75005292333235060782295002381, 0,
0.75005292333235060782295002381, 3.04353762371028671316761287882, 4.72814705073561007578831702072, 6.50213031735933206148033533821, 7.24704027093279717365420605400, 8.447693182029751111158905221001, 9.432705067809431215474966666216, 10.56347601143106942416278207090, 11.34880910121064609283660591817
