| L(s) = 1 | − 28.5·2-s + 729·3-s − 7.37e3·4-s − 1.38e4·5-s − 2.07e4·6-s + 3.51e5·7-s + 4.43e5·8-s + 5.31e5·9-s + 3.95e5·10-s + 8.63e6·11-s − 5.37e6·12-s − 2.28e7·13-s − 1.00e7·14-s − 1.01e7·15-s + 4.77e7·16-s − 1.05e6·17-s − 1.51e7·18-s − 3.21e7·19-s + 1.02e8·20-s + 2.56e8·21-s − 2.46e8·22-s − 5.53e8·23-s + 3.23e8·24-s − 1.02e9·25-s + 6.52e8·26-s + 3.87e8·27-s − 2.59e9·28-s + ⋯ |
| L(s) = 1 | − 0.314·2-s + 0.577·3-s − 0.900·4-s − 0.397·5-s − 0.181·6-s + 1.12·7-s + 0.598·8-s + 0.333·9-s + 0.125·10-s + 1.46·11-s − 0.520·12-s − 1.31·13-s − 0.355·14-s − 0.229·15-s + 0.712·16-s − 0.0105·17-s − 0.104·18-s − 0.156·19-s + 0.358·20-s + 0.652·21-s − 0.462·22-s − 0.779·23-s + 0.345·24-s − 0.842·25-s + 0.414·26-s + 0.192·27-s − 1.01·28-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 | \( 1 + 28.5T + 8.19e3T^{2} \) |
| 5 | \( 1 + 1.38e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.51e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 8.63e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.28e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.05e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.21e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 5.53e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.88e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.59e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.31e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.93e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 7.30e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 7.35e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.27e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.72e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.83e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 3.93e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.61e10T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.21e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.98e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.57e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.94e12T + 6.73e25T^{2} \) |
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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
−9.506488669571070519457884637071, −8.949192141651261678678141466653, −7.905227999123865911867737973841, −7.28163909109363095851764693271, −5.55550331634386407861107646825, −4.35823662326415612673633772144, −3.87249759211717344715550965344, −2.16406220079347355969764904033, −1.20862746197247855056610354277, 0,
1.20862746197247855056610354277, 2.16406220079347355969764904033, 3.87249759211717344715550965344, 4.35823662326415612673633772144, 5.55550331634386407861107646825, 7.28163909109363095851764693271, 7.905227999123865911867737973841, 8.949192141651261678678141466653, 9.506488669571070519457884637071
