| L(s) = 1 | + 20.7·3-s − 501.·5-s − 1.00e3·7-s − 1.75e3·9-s + 6.78e3·11-s + 1.02e4·13-s − 1.03e4·15-s + 2.78e4·17-s + 6.85e3·19-s − 2.09e4·21-s + 4.35e3·23-s + 1.73e5·25-s − 8.17e4·27-s − 1.71e5·29-s − 1.07e5·31-s + 1.40e5·33-s + 5.05e5·35-s + 4.89e5·37-s + 2.12e5·39-s + 2.59e4·41-s + 1.89e5·43-s + 8.80e5·45-s − 4.04e5·47-s + 1.92e5·49-s + 5.77e5·51-s + 1.48e6·53-s − 3.40e6·55-s + ⋯ |
| L(s) = 1 | + 0.443·3-s − 1.79·5-s − 1.11·7-s − 0.803·9-s + 1.53·11-s + 1.29·13-s − 0.795·15-s + 1.37·17-s + 0.229·19-s − 0.492·21-s + 0.0745·23-s + 2.21·25-s − 0.799·27-s − 1.30·29-s − 0.645·31-s + 0.681·33-s + 1.99·35-s + 1.58·37-s + 0.573·39-s + 0.0587·41-s + 0.363·43-s + 1.44·45-s − 0.568·47-s + 0.233·49-s + 0.609·51-s + 1.36·53-s − 2.75·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.359398761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359398761\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 - 20.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 501.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.00e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.78e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.02e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.78e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 4.35e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.71e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.89e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.59e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.89e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.04e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.48e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.34e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.44e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.99e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.45e7T + 8.07e13T^{2} \) |
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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
−12.96092492628838036557267327373, −11.84164094138381605325484526712, −11.19599159845887946454176349245, −9.423487801949890719866612780021, −8.482709290860602908621699207172, −7.38301485754590710144456040696, −6.04694164979073317308391147589, −3.79801490038615456046475074895, −3.39095682141689116028038862073, −0.76093887302894872095068258384,
0.76093887302894872095068258384, 3.39095682141689116028038862073, 3.79801490038615456046475074895, 6.04694164979073317308391147589, 7.38301485754590710144456040696, 8.482709290860602908621699207172, 9.423487801949890719866612780021, 11.19599159845887946454176349245, 11.84164094138381605325484526712, 12.96092492628838036557267327373
