| L(s) = 1 | − 8·2-s + 89.1·3-s + 64·4-s + 535.·5-s − 713.·6-s + 343·7-s − 512·8-s + 5.76e3·9-s − 4.28e3·10-s − 4.01e3·11-s + 5.70e3·12-s + 2.19e3·13-s − 2.74e3·14-s + 4.77e4·15-s + 4.09e3·16-s − 2.80e4·17-s − 4.61e4·18-s + 7.31e3·19-s + 3.42e4·20-s + 3.05e4·21-s + 3.21e4·22-s + 4.96e4·23-s − 4.56e4·24-s + 2.08e5·25-s − 1.75e4·26-s + 3.19e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s + 1.91·5-s − 1.34·6-s + 0.377·7-s − 0.353·8-s + 2.63·9-s − 1.35·10-s − 0.910·11-s + 0.953·12-s + 0.277·13-s − 0.267·14-s + 3.65·15-s + 0.250·16-s − 1.38·17-s − 1.86·18-s + 0.244·19-s + 0.957·20-s + 0.720·21-s + 0.643·22-s + 0.851·23-s − 0.674·24-s + 2.66·25-s − 0.196·26-s + 3.12·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.708577932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.708577932\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 3 | \( 1 - 89.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 535.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.01e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.80e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 7.31e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.94e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.92e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.26e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.67e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.05e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.26e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.29e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.34e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.06e6T + 8.07e13T^{2} \) |
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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
−10.74075340385203157096994272848, −10.03385159923627085444664940487, −9.130388808771143029719475093392, −8.683746626537078446784024059705, −7.52220013058632878747073973366, −6.42540528699035434425200918852, −4.85452475222294541435708649345, −2.95839161641528972475786310957, −2.20983934495279300897652769973, −1.41231190632289482085141569789,
1.41231190632289482085141569789, 2.20983934495279300897652769973, 2.95839161641528972475786310957, 4.85452475222294541435708649345, 6.42540528699035434425200918852, 7.52220013058632878747073973366, 8.683746626537078446784024059705, 9.130388808771143029719475093392, 10.03385159923627085444664940487, 10.74075340385203157096994272848
