| L(s) = 1 | − 3.32·2-s − 243·3-s − 2.03e3·4-s + 1.03e4·5-s + 807.·6-s + 3.60e4·7-s + 1.35e4·8-s + 5.90e4·9-s − 3.45e4·10-s + 2.10e5·11-s + 4.94e5·12-s − 1.65e5·13-s − 1.19e5·14-s − 2.52e6·15-s + 4.12e6·16-s + 1.10e7·17-s − 1.96e5·18-s + 6.36e6·19-s − 2.11e7·20-s − 8.74e6·21-s − 6.99e5·22-s + 3.65e7·23-s − 3.29e6·24-s + 5.90e7·25-s + 5.49e5·26-s − 1.43e7·27-s − 7.33e7·28-s + ⋯ |
| L(s) = 1 | − 0.0734·2-s − 0.577·3-s − 0.994·4-s + 1.48·5-s + 0.0423·6-s + 0.809·7-s + 0.146·8-s + 0.333·9-s − 0.109·10-s + 0.394·11-s + 0.574·12-s − 0.123·13-s − 0.0594·14-s − 0.858·15-s + 0.983·16-s + 1.89·17-s − 0.0244·18-s + 0.589·19-s − 1.47·20-s − 0.467·21-s − 0.0289·22-s + 1.18·23-s − 0.0845·24-s + 1.21·25-s + 0.00906·26-s − 0.192·27-s − 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.720714540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.720714540\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 + 3.32T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.03e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.60e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.10e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.65e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.10e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.36e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.65e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 4.69e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.92e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 8.53e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.98e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.07e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 4.33e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.70e7T + 9.26e18T^{2} \) |
| 61 | \( 1 + 7.10e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.47e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.78e8T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.94e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.66e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.27e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.31e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.93e10T + 7.15e21T^{2} \) |
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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
−10.29625094205070099711785964814, −9.842533350674839084416506554871, −8.843997271989220040786685929988, −7.70710396216186325528217113277, −6.27357426273313093675091103583, −5.31819195831311908461031894211, −4.76540239250023413185274407857, −3.14176556838132930471202946149, −1.47609094327459646299838759636, −0.916898562946390714450685344364,
0.916898562946390714450685344364, 1.47609094327459646299838759636, 3.14176556838132930471202946149, 4.76540239250023413185274407857, 5.31819195831311908461031894211, 6.27357426273313093675091103583, 7.70710396216186325528217113277, 8.843997271989220040786685929988, 9.842533350674839084416506554871, 10.29625094205070099711785964814
