| L(s) = 1 | + 2.64·2-s + 25.8·3-s − 121.·4-s + 466.·5-s + 68.3·6-s − 343·7-s − 658.·8-s − 1.52e3·9-s + 1.23e3·10-s − 7.79e3·11-s − 3.12e3·12-s + 2.19e3·13-s − 907.·14-s + 1.20e4·15-s + 1.37e4·16-s − 1.30e4·17-s − 4.02e3·18-s + 4.10e4·19-s − 5.64e4·20-s − 8.85e3·21-s − 2.06e4·22-s − 5.66e4·23-s − 1.70e4·24-s + 1.39e5·25-s + 5.81e3·26-s − 9.57e4·27-s + 4.15e4·28-s + ⋯ |
| L(s) = 1 | + 0.233·2-s + 0.552·3-s − 0.945·4-s + 1.66·5-s + 0.129·6-s − 0.377·7-s − 0.454·8-s − 0.695·9-s + 0.390·10-s − 1.76·11-s − 0.522·12-s + 0.277·13-s − 0.0883·14-s + 0.921·15-s + 0.838·16-s − 0.643·17-s − 0.162·18-s + 1.37·19-s − 1.57·20-s − 0.208·21-s − 0.412·22-s − 0.971·23-s − 0.251·24-s + 1.78·25-s + 0.0648·26-s − 0.936·27-s + 0.357·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 2.64T + 128T^{2} \) |
| 3 | \( 1 - 25.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 466.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.79e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.30e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.66e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.39e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.66e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.10e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.29e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.60e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.85e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.95e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.42e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.96e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.82e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.05e6T + 8.07e13T^{2} \) |
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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
−12.68943786357204575500849272395, −10.78702055228626314189324495718, −9.653082989062852073747801678264, −9.089147681302841983639211528115, −7.79171292629036609137153696453, −5.81963611630159241657119439116, −5.28554330339411357633182240150, −3.30535359078773361311988378547, −2.10231416428804268686953780943, 0,
2.10231416428804268686953780943, 3.30535359078773361311988378547, 5.28554330339411357633182240150, 5.81963611630159241657119439116, 7.79171292629036609137153696453, 9.089147681302841983639211528115, 9.653082989062852073747801678264, 10.78702055228626314189324495718, 12.68943786357204575500849272395
