| L(s) = 1 | − 9.60·2-s + 27·3-s − 35.7·4-s + 26.4·5-s − 259.·6-s − 248.·7-s + 1.57e3·8-s + 729·9-s − 253.·10-s + 43.2·11-s − 964.·12-s − 5.82e3·13-s + 2.38e3·14-s + 713.·15-s − 1.05e4·16-s + 4.64e3·17-s − 7.00e3·18-s + 4.54e4·19-s − 944.·20-s − 6.69e3·21-s − 415.·22-s − 1.21e4·23-s + 4.24e4·24-s − 7.74e4·25-s + 5.59e4·26-s + 1.96e4·27-s + 8.86e3·28-s + ⋯ |
| L(s) = 1 | − 0.849·2-s + 0.577·3-s − 0.278·4-s + 0.0945·5-s − 0.490·6-s − 0.273·7-s + 1.08·8-s + 0.333·9-s − 0.0803·10-s + 0.00980·11-s − 0.161·12-s − 0.735·13-s + 0.232·14-s + 0.0546·15-s − 0.643·16-s + 0.229·17-s − 0.283·18-s + 1.51·19-s − 0.0263·20-s − 0.157·21-s − 0.00832·22-s − 0.208·23-s + 0.627·24-s − 0.991·25-s + 0.624·26-s + 0.192·27-s + 0.0762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 9.60T + 128T^{2} \) |
| 5 | \( 1 - 26.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + 248.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 43.2T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.82e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.64e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.54e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.57e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.42e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.28e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.15e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.75e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.84e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.59e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.48e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.32e7T + 8.07e13T^{2} \) |
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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
−12.92284424367205736078011300152, −11.44083596361352550702532693072, −9.864678413740451516006604135412, −9.482551713779357653882228630185, −8.120862782850411435717036575124, −7.22845944127439019869698153405, −5.23477670095534902872271640838, −3.57733097162738005483956596242, −1.71816092815786947279806552333, 0,
1.71816092815786947279806552333, 3.57733097162738005483956596242, 5.23477670095534902872271640838, 7.22845944127439019869698153405, 8.120862782850411435717036575124, 9.482551713779357653882228630185, 9.864678413740451516006604135412, 11.44083596361352550702532693072, 12.92284424367205736078011300152
