| L(s) = 1 | + 52.9·2-s − 729·3-s − 5.38e3·4-s − 3.31e4·5-s − 3.85e4·6-s − 4.13e5·7-s − 7.18e5·8-s + 5.31e5·9-s − 1.75e6·10-s + 2.37e6·11-s + 3.92e6·12-s − 1.44e7·13-s − 2.18e7·14-s + 2.41e7·15-s + 6.09e6·16-s − 9.80e7·17-s + 2.81e7·18-s + 2.31e8·19-s + 1.78e8·20-s + 3.01e8·21-s + 1.25e8·22-s + 7.61e8·23-s + 5.24e8·24-s − 1.21e8·25-s − 7.66e8·26-s − 3.87e8·27-s + 2.22e9·28-s + ⋯ |
| L(s) = 1 | + 0.584·2-s − 0.577·3-s − 0.657·4-s − 0.948·5-s − 0.337·6-s − 1.32·7-s − 0.969·8-s + 0.333·9-s − 0.554·10-s + 0.403·11-s + 0.379·12-s − 0.832·13-s − 0.776·14-s + 0.547·15-s + 0.0908·16-s − 0.985·17-s + 0.194·18-s + 1.13·19-s + 0.624·20-s + 0.766·21-s + 0.235·22-s + 1.07·23-s + 0.559·24-s − 0.0997·25-s − 0.486·26-s − 0.192·27-s + 0.873·28-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
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 52.9T + 8.19e3T^{2} \) |
| 5 | \( 1 + 3.31e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 4.13e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 2.37e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.44e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 9.80e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.31e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 7.61e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.21e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.65e8T + 2.44e19T^{2} \) |
| 37 | \( 1 - 3.56e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 5.94e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.00e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 7.04e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.38e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 4.48e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.96e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 4.00e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.04e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.96e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.20e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.35e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 6.74e12T + 6.73e25T^{2} \) |
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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
−9.652588184537143780420006869217, −9.139614166881172271222515904447, −7.62419080476729346996757796679, −6.67033404598368493105382215259, −5.62784266809255973450324763293, −4.55336437096867093311163353856, −3.74818627487371735768268992963, −2.80172076546955243393319066483, −0.75000679266519900201010592475, 0,
0.75000679266519900201010592475, 2.80172076546955243393319066483, 3.74818627487371735768268992963, 4.55336437096867093311163353856, 5.62784266809255973450324763293, 6.67033404598368493105382215259, 7.62419080476729346996757796679, 9.139614166881172271222515904447, 9.652588184537143780420006869217
