| L(s) = 1 | + (32.9 + 136. i)3-s − 119.·5-s + 9.28e3i·7-s + (−1.75e4 + 8.98e3i)9-s − 2.44e4i·11-s − 7.49e4i·13-s + (−3.92e3 − 1.62e4i)15-s + 5.84e5i·17-s + 1.17e5·19-s + (−1.26e6 + 3.06e5i)21-s − 1.43e6·23-s − 1.93e6·25-s + (−1.80e6 − 2.09e6i)27-s + 1.18e6·29-s − 3.45e6i·31-s + ⋯ |
| L(s) = 1 | + (0.234 + 0.972i)3-s − 0.0851·5-s + 1.46i·7-s + (−0.889 + 0.456i)9-s − 0.503i·11-s − 0.727i·13-s + (−0.0200 − 0.0828i)15-s + 1.69i·17-s + 0.206·19-s + (−1.42 + 0.343i)21-s − 1.06·23-s − 0.992·25-s + (−0.652 − 0.757i)27-s + 0.312·29-s − 0.672i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0246 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0246 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1754923394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1754923394\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-32.9 - 136. i)T \) |
| good | 5 | \( 1 + 119.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.28e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 2.44e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 7.49e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 5.84e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 1.17e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.43e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.45e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.67e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.37e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 7.60e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.29e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.69e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.29e8iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 2.02e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 7.41e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.02e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.86e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.77e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 5.24e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 4.74e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 - 4.95e8T + 7.60e17T^{2} \) |
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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
−11.56528828687807014888701002090, −10.49429659963538125644800905240, −9.669529399222321364913008450778, −8.547447829069969532110897806082, −8.102586026668057010405346287241, −6.02571471599253810818914559011, −5.54065560224526182622023674326, −4.12597909005583017254842122539, −3.08011760707855850282695756412, −1.96687003105821421913065230036,
0.03876322023910695232314130205, 1.03996846575588023720719818579, 2.18734884151407634902686724897, 3.56789351623890692575759243004, 4.72627088199947204023014423241, 6.28879674469191703587668530008, 7.28014025357625443303396335209, 7.70674777622609891933842569020, 9.134175812223581724139772367476, 10.08141355508422940683868624931
