L(s) = 1 | + 9.74·2-s + 27·3-s − 33.0·4-s + 321.·5-s + 263.·6-s − 939.·7-s − 1.56e3·8-s + 729·9-s + 3.13e3·10-s + 4.99e3·11-s − 892.·12-s − 1.23e3·13-s − 9.15e3·14-s + 8.69e3·15-s − 1.10e4·16-s + 9.78e3·17-s + 7.10e3·18-s + 1.93e4·19-s − 1.06e4·20-s − 2.53e4·21-s + 4.87e4·22-s + 7.98e4·23-s − 4.23e4·24-s + 2.54e4·25-s − 1.20e4·26-s + 1.96e4·27-s + 3.10e4·28-s + ⋯ |
L(s) = 1 | + 0.861·2-s + 0.577·3-s − 0.258·4-s + 1.15·5-s + 0.497·6-s − 1.03·7-s − 1.08·8-s + 0.333·9-s + 0.991·10-s + 1.13·11-s − 0.149·12-s − 0.155·13-s − 0.892·14-s + 0.664·15-s − 0.675·16-s + 0.482·17-s + 0.287·18-s + 0.646·19-s − 0.297·20-s − 0.597·21-s + 0.975·22-s + 1.36·23-s − 0.625·24-s + 0.326·25-s − 0.134·26-s + 0.192·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(4.259980291\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.259980291\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 - 9.74T + 128T^{2} \) |
| 5 | \( 1 - 321.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 939.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 9.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.93e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.48e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.03e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.43e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.86e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.28e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.54e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.62e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.93e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.86e6T + 8.07e13T^{2} \) |
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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
−11.75331271740602004033339582783, −9.969038730848036175223439052155, −9.519890997480776203432502710553, −8.633928776022739491142023550168, −6.83147603286064007276315882041, −6.05491173321406562231315089982, −4.89417946934239545579347695306, −3.56858170228550914446705659247, −2.70236902239834437967691797107, −1.02844007919417945031662637728,
1.02844007919417945031662637728, 2.70236902239834437967691797107, 3.56858170228550914446705659247, 4.89417946934239545579347695306, 6.05491173321406562231315089982, 6.83147603286064007276315882041, 8.633928776022739491142023550168, 9.519890997480776203432502710553, 9.969038730848036175223439052155, 11.75331271740602004033339582783