| L(s) = 1 | − 14.6·2-s − 51.4·3-s + 85.7·4-s + 123.·5-s + 752.·6-s + 559.·7-s + 618.·8-s + 459.·9-s − 1.80e3·10-s + 1.82e3·11-s − 4.40e3·12-s − 2.19e3·13-s − 8.18e3·14-s − 6.36e3·15-s − 2.00e4·16-s + 3.36e4·17-s − 6.71e3·18-s + 4.26e4·19-s + 1.05e4·20-s − 2.87e4·21-s − 2.66e4·22-s + 5.53e4·23-s − 3.17e4·24-s − 6.28e4·25-s + 3.21e4·26-s + 8.88e4·27-s + 4.79e4·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 1.09·3-s + 0.669·4-s + 0.442·5-s + 1.42·6-s + 0.616·7-s + 0.426·8-s + 0.209·9-s − 0.571·10-s + 0.412·11-s − 0.736·12-s − 0.277·13-s − 0.796·14-s − 0.486·15-s − 1.22·16-s + 1.66·17-s − 0.271·18-s + 1.42·19-s + 0.296·20-s − 0.678·21-s − 0.533·22-s + 0.949·23-s − 0.469·24-s − 0.804·25-s + 0.358·26-s + 0.869·27-s + 0.412·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5921082170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5921082170\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 14.6T + 128T^{2} \) |
| 3 | \( 1 + 51.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 123.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 559.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.82e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.53e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.84e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.42e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.06e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.49e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.79e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.44e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.84e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.49e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.29e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.03e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.12e7T + 8.07e13T^{2} \) |
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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
−17.92725489934734562116919788143, −17.25022111419264492745432766356, −16.29858667278964127414775829004, −14.15104045479500406400665344606, −11.97048879922388568343575913691, −10.71939130037177633192534161685, −9.376029207733058478307222521551, −7.52144814924429281070011042230, −5.43373787335579278131301623102, −1.05036531994160171881272046292,
1.05036531994160171881272046292, 5.43373787335579278131301623102, 7.52144814924429281070011042230, 9.376029207733058478307222521551, 10.71939130037177633192534161685, 11.97048879922388568343575913691, 14.15104045479500406400665344606, 16.29858667278964127414775829004, 17.25022111419264492745432766356, 17.92725489934734562116919788143
