| L(s) = 1 | − 22.2·2-s − 729·3-s − 7.69e3·4-s − 2.95e4·5-s + 1.62e4·6-s − 2.11e5·7-s + 3.54e5·8-s + 5.31e5·9-s + 6.59e5·10-s + 6.46e5·11-s + 5.60e6·12-s + 1.28e7·13-s + 4.70e6·14-s + 2.15e7·15-s + 5.51e7·16-s + 1.59e8·17-s − 1.18e7·18-s + 2.48e8·19-s + 2.27e8·20-s + 1.53e8·21-s − 1.44e7·22-s − 9.45e8·23-s − 2.58e8·24-s − 3.45e8·25-s − 2.87e8·26-s − 3.87e8·27-s + 1.62e9·28-s + ⋯ |
| L(s) = 1 | − 0.246·2-s − 0.577·3-s − 0.939·4-s − 0.846·5-s + 0.142·6-s − 0.678·7-s + 0.477·8-s + 0.333·9-s + 0.208·10-s + 0.110·11-s + 0.542·12-s + 0.741·13-s + 0.167·14-s + 0.488·15-s + 0.821·16-s + 1.60·17-s − 0.0821·18-s + 1.21·19-s + 0.795·20-s + 0.391·21-s − 0.0271·22-s − 1.33·23-s − 0.275·24-s − 0.282·25-s − 0.182·26-s − 0.192·27-s + 0.637·28-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
| \(L(7)\) |
\(\approx\) |
\(0.6966148418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6966148418\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 22.2T + 8.19e3T^{2} \) |
| 5 | \( 1 + 2.95e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.11e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 6.46e5T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.28e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.59e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.48e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 9.45e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.51e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 8.29e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 6.67e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.93e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 6.65e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.24e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.12e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 7.41e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 9.43e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 6.15e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.10e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.09e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.10e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.41e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.09e13T + 6.73e25T^{2} \) |
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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
−10.04511189677105669001193165977, −9.609952619058810411120962170822, −8.198520199979263175436268851302, −7.59410391725084456551720685277, −6.14987593613340069593456879365, −5.24241859299591286850282199324, −3.97515041343567697349614642209, −3.37582762337459394355053922248, −1.31564921837581657270514162887, −0.43324353657440888686376699258,
0.43324353657440888686376699258, 1.31564921837581657270514162887, 3.37582762337459394355053922248, 3.97515041343567697349614642209, 5.24241859299591286850282199324, 6.14987593613340069593456879365, 7.59410391725084456551720685277, 8.198520199979263175436268851302, 9.609952619058810411120962170822, 10.04511189677105669001193165977
