| L(s) = 1 | − 112.·2-s + 729·3-s + 4.54e3·4-s + 3.76e4·5-s − 8.22e4·6-s − 6.04e4·7-s + 4.11e5·8-s + 5.31e5·9-s − 4.24e6·10-s − 5.38e6·11-s + 3.31e6·12-s + 1.78e7·13-s + 6.82e6·14-s + 2.74e7·15-s − 8.36e7·16-s − 7.26e7·17-s − 5.99e7·18-s + 2.05e8·19-s + 1.71e8·20-s − 4.40e7·21-s + 6.07e8·22-s − 2.11e8·23-s + 2.99e8·24-s + 1.93e8·25-s − 2.01e9·26-s + 3.87e8·27-s − 2.75e8·28-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.577·3-s + 0.555·4-s + 1.07·5-s − 0.720·6-s − 0.194·7-s + 0.554·8-s + 0.333·9-s − 1.34·10-s − 0.916·11-s + 0.320·12-s + 1.02·13-s + 0.242·14-s + 0.621·15-s − 1.24·16-s − 0.730·17-s − 0.415·18-s + 1.00·19-s + 0.597·20-s − 0.112·21-s + 1.14·22-s − 0.298·23-s + 0.320·24-s + 0.158·25-s − 1.27·26-s + 0.192·27-s − 0.107·28-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 | \( 1 + 112.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.76e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 6.04e4T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.38e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.78e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 7.26e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.05e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 2.11e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.90e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.76e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 5.09e8T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.36e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.35e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.26e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.89e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 2.30e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.70e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.48e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.87e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.52e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 1.34e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 1.41e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.48e13T + 6.73e25T^{2} \) |
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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
−9.788511651241524618112570133389, −8.884940668368815404950042348044, −8.182210997847131772634082475486, −7.11458093920980395998755055237, −5.97645668553867525723310364092, −4.70838414644884237389245645202, −3.15545658331977947525732615051, −2.02452971237693917649156158407, −1.25022662048532349888933592876, 0,
1.25022662048532349888933592876, 2.02452971237693917649156158407, 3.15545658331977947525732615051, 4.70838414644884237389245645202, 5.97645668553867525723310364092, 7.11458093920980395998755055237, 8.182210997847131772634082475486, 8.884940668368815404950042348044, 9.788511651241524618112570133389
