| L(s) = 1 | − 6.98·2-s − 729·3-s − 8.14e3·4-s − 4.31e3·5-s + 5.09e3·6-s − 1.09e4·7-s + 1.14e5·8-s + 5.31e5·9-s + 3.01e4·10-s + 5.41e5·11-s + 5.93e6·12-s − 1.28e7·13-s + 7.64e4·14-s + 3.14e6·15-s + 6.59e7·16-s + 4.12e7·17-s − 3.71e6·18-s + 6.07e7·19-s + 3.51e7·20-s + 7.97e6·21-s − 3.78e6·22-s + 2.01e7·23-s − 8.32e7·24-s − 1.20e9·25-s + 9.01e7·26-s − 3.87e8·27-s + 8.91e7·28-s + ⋯ |
| L(s) = 1 | − 0.0771·2-s − 0.577·3-s − 0.994·4-s − 0.123·5-s + 0.0445·6-s − 0.0351·7-s + 0.153·8-s + 0.333·9-s + 0.00953·10-s + 0.0922·11-s + 0.573·12-s − 0.740·13-s + 0.00271·14-s + 0.0712·15-s + 0.982·16-s + 0.414·17-s − 0.0257·18-s + 0.296·19-s + 0.122·20-s + 0.0202·21-s − 0.00711·22-s + 0.0284·23-s − 0.0888·24-s − 0.984·25-s + 0.0572·26-s − 0.192·27-s + 0.0349·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.7310986081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7310986081\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 + 6.98T + 8.19e3T^{2} \) |
| 5 | \( 1 + 4.31e3T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.09e4T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.41e5T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.28e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 4.12e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 6.07e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.01e7T + 5.04e17T^{2} \) |
| 29 | \( 1 - 6.14e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 6.53e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.85e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.49e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 8.13e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.94e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.61e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 6.17e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 8.07e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.46e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.41e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 7.71e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 6.75e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.58e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 7.60e12T + 6.73e25T^{2} \) |
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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
−10.11255138283704319287964681072, −9.554772086118095342742417217494, −8.343016031098920500972434492483, −7.42621810018408092695661068639, −6.13234510815492364865431390827, −5.10096355312923773184406588842, −4.31995379674220583824323517553, −3.11972843182918065398245787369, −1.50390153882161573705388598984, −0.40009833240204915415642382004,
0.40009833240204915415642382004, 1.50390153882161573705388598984, 3.11972843182918065398245787369, 4.31995379674220583824323517553, 5.10096355312923773184406588842, 6.13234510815492364865431390827, 7.42621810018408092695661068639, 8.343016031098920500972434492483, 9.554772086118095342742417217494, 10.11255138283704319287964681072
