| L(s) = 1 | − 21.0·2-s − 33.4·3-s + 315.·4-s + 271.·5-s + 703.·6-s − 343·7-s − 3.94e3·8-s − 1.06e3·9-s − 5.71e3·10-s − 610.·11-s − 1.05e4·12-s + 2.19e3·13-s + 7.22e3·14-s − 9.07e3·15-s + 4.26e4·16-s + 8.45e3·17-s + 2.25e4·18-s + 3.64e3·19-s + 8.55e4·20-s + 1.14e4·21-s + 1.28e4·22-s + 2.90e4·23-s + 1.31e5·24-s − 4.44e3·25-s − 4.62e4·26-s + 1.08e5·27-s − 1.08e5·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.714·3-s + 2.46·4-s + 0.971·5-s + 1.33·6-s − 0.377·7-s − 2.72·8-s − 0.488·9-s − 1.80·10-s − 0.138·11-s − 1.76·12-s + 0.277·13-s + 0.703·14-s − 0.694·15-s + 2.60·16-s + 0.417·17-s + 0.909·18-s + 0.121·19-s + 2.39·20-s + 0.270·21-s + 0.257·22-s + 0.497·23-s + 1.94·24-s − 0.0568·25-s − 0.516·26-s + 1.06·27-s − 0.930·28-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 | \( 1 + 21.0T + 128T^{2} \) |
| 3 | \( 1 + 33.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 271.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 610.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 8.45e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.64e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.90e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.53e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.65e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.65e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.46e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.61e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.05e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.17e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.49e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.71e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.09e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.31e7T + 8.07e13T^{2} \) |
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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
−11.57528607109332780056044621367, −10.80326614138835858328810439287, −9.814316479780883407193343798408, −9.066635895813777337279876414150, −7.79432062261302116620143933330, −6.47064081637744625272608618827, −5.67722785789557672405028268223, −2.72551537988363500379532921733, −1.29747157780935328742476209569, 0,
1.29747157780935328742476209569, 2.72551537988363500379532921733, 5.67722785789557672405028268223, 6.47064081637744625272608618827, 7.79432062261302116620143933330, 9.066635895813777337279876414150, 9.814316479780883407193343798408, 10.80326614138835858328810439287, 11.57528607109332780056044621367
