| L(s) = 1 | − 83.5·2-s + 729·3-s − 1.20e3·4-s + 5.58e4·5-s − 6.09e4·6-s − 5.15e5·7-s + 7.85e5·8-s + 5.31e5·9-s − 4.66e6·10-s + 3.72e6·11-s − 8.81e5·12-s − 9.32e6·13-s + 4.30e7·14-s + 4.07e7·15-s − 5.57e7·16-s − 7.95e6·17-s − 4.44e7·18-s − 6.13e7·19-s − 6.75e7·20-s − 3.75e8·21-s − 3.11e8·22-s + 5.74e8·23-s + 5.72e8·24-s + 1.89e9·25-s + 7.79e8·26-s + 3.87e8·27-s + 6.23e8·28-s + ⋯ |
| L(s) = 1 | − 0.923·2-s + 0.577·3-s − 0.147·4-s + 1.59·5-s − 0.533·6-s − 1.65·7-s + 1.05·8-s + 0.333·9-s − 1.47·10-s + 0.633·11-s − 0.0852·12-s − 0.536·13-s + 1.52·14-s + 0.923·15-s − 0.830·16-s − 0.0799·17-s − 0.307·18-s − 0.299·19-s − 0.235·20-s − 0.955·21-s − 0.585·22-s + 0.808·23-s + 0.611·24-s + 1.55·25-s + 0.494·26-s + 0.192·27-s + 0.244·28-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
| \(L(7)\) |
\(\approx\) |
\(1.582494131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582494131\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 83.5T + 8.19e3T^{2} \) |
| 5 | \( 1 - 5.58e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.15e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 3.72e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 9.32e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 7.95e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 6.13e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.74e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.68e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.08e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.19e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 7.06e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.21e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 3.35e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.04e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.12e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 5.18e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 7.67e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 6.21e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.09e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 8.75e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.59e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 4.07e12T + 6.73e25T^{2} \) |
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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
−9.915836245891466965410099258128, −9.256246443364565837584990550647, −8.987754345374828074015575664649, −7.33891818419316509333527965740, −6.50580496763658497508188438617, −5.40620223027287071728447935660, −3.90072919448995326965508281797, −2.66117780092369346150963099598, −1.72196746474144329079539193319, −0.60061723644905267186066281175,
0.60061723644905267186066281175, 1.72196746474144329079539193319, 2.66117780092369346150963099598, 3.90072919448995326965508281797, 5.40620223027287071728447935660, 6.50580496763658497508188438617, 7.33891818419316509333527965740, 8.987754345374828074015575664649, 9.256246443364565837584990550647, 9.915836245891466965410099258128
