| L(s) = 1 | − 11.5·2-s + 23.9·3-s + 4.51·4-s − 301.·5-s − 275.·6-s + 343·7-s + 1.42e3·8-s − 1.61e3·9-s + 3.46e3·10-s + 8.16e3·11-s + 107.·12-s − 2.19e3·13-s − 3.94e3·14-s − 7.20e3·15-s − 1.69e4·16-s − 1.84e4·17-s + 1.85e4·18-s + 2.54e4·19-s − 1.35e3·20-s + 8.19e3·21-s − 9.39e4·22-s + 1.00e5·23-s + 3.39e4·24-s + 1.27e4·25-s + 2.52e4·26-s − 9.09e4·27-s + 1.54e3·28-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.511·3-s + 0.0352·4-s − 1.07·5-s − 0.520·6-s + 0.377·7-s + 0.981·8-s − 0.738·9-s + 1.09·10-s + 1.84·11-s + 0.0180·12-s − 0.277·13-s − 0.384·14-s − 0.551·15-s − 1.03·16-s − 0.910·17-s + 0.751·18-s + 0.851·19-s − 0.0380·20-s + 0.193·21-s − 1.88·22-s + 1.72·23-s + 0.501·24-s + 0.162·25-s + 0.282·26-s − 0.888·27-s + 0.0133·28-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 | \( 1 + 11.5T + 128T^{2} \) |
| 3 | \( 1 - 23.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 301.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 8.16e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.54e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.00e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.63e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.83e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.64e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.59e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.10e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.22e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.66e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.83e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.85e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.84e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.74e6T + 8.07e13T^{2} \) |
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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
−11.64769916107716751134888488552, −11.21319982772869122339618769685, −9.473291860409318164752372736481, −8.834539130154347333895689680784, −7.934065153265051592763615413458, −6.86116249363622835425109262823, −4.70330233272987203415396040068, −3.44173239824496406507770334297, −1.43044142206764317448831476794, 0,
1.43044142206764317448831476794, 3.44173239824496406507770334297, 4.70330233272987203415396040068, 6.86116249363622835425109262823, 7.934065153265051592763615413458, 8.834539130154347333895689680784, 9.473291860409318164752372736481, 11.21319982772869122339618769685, 11.64769916107716751134888488552
