| L(s) = 1 | − 67.4·2-s + 243·3-s + 2.50e3·4-s − 169.·5-s − 1.63e4·6-s + 6.57e4·7-s − 3.07e4·8-s + 5.90e4·9-s + 1.14e4·10-s − 8.51e5·11-s + 6.08e5·12-s − 6.45e4·13-s − 4.43e6·14-s − 4.11e4·15-s − 3.05e6·16-s + 4.48e6·17-s − 3.98e6·18-s − 8.64e6·19-s − 4.24e5·20-s + 1.59e7·21-s + 5.74e7·22-s − 8.74e6·23-s − 7.47e6·24-s − 4.87e7·25-s + 4.35e6·26-s + 1.43e7·27-s + 1.64e8·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.22·4-s − 0.0242·5-s − 0.860·6-s + 1.47·7-s − 0.331·8-s + 0.333·9-s + 0.0361·10-s − 1.59·11-s + 0.705·12-s − 0.0482·13-s − 2.20·14-s − 0.0140·15-s − 0.727·16-s + 0.765·17-s − 0.496·18-s − 0.800·19-s − 0.0296·20-s + 0.853·21-s + 2.37·22-s − 0.283·23-s − 0.191·24-s − 0.999·25-s + 0.0719·26-s + 0.192·27-s + 1.80·28-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
| \(L(6)\) |
\(\approx\) |
\(1.257117332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.257117332\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 67.4T + 2.04e3T^{2} \) |
| 5 | \( 1 + 169.T + 4.88e7T^{2} \) |
| 7 | \( 1 - 6.57e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.51e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 6.45e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.48e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.64e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 8.74e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.62e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.58e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.12e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.09e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.73e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.85e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.57e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.00e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.14e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.70e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.15e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 5.14e8T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.94e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 6.81e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.84e10T + 7.15e21T^{2} \) |
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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
−10.53064988556413975958746435434, −9.590242923332630636005324979287, −8.396001599482433578473127872623, −8.010226445193033399578515840315, −7.28186792541692363039317745729, −5.50463479604220806666768657454, −4.31596398969455363727299753429, −2.52614358278414961161624312084, −1.77302296662823510489516392548, −0.62352861414663876489159609705,
0.62352861414663876489159609705, 1.77302296662823510489516392548, 2.52614358278414961161624312084, 4.31596398969455363727299753429, 5.50463479604220806666768657454, 7.28186792541692363039317745729, 8.010226445193033399578515840315, 8.396001599482433578473127872623, 9.590242923332630636005324979287, 10.53064988556413975958746435434
