| L(s) = 1 | − 30.9·3-s − 10i·5-s + 309. i·7-s + 231.·9-s + 960.·11-s + 1.46e3i·13-s + 309. i·15-s − 4.76e3·17-s + 7.52e3·19-s − 9.60e3i·21-s − 1.04e4i·23-s + 1.55e4·25-s + 1.54e4·27-s + 2.54e4i·29-s − 4.18e4i·31-s + ⋯ |
| L(s) = 1 | − 1.14·3-s − 0.0800i·5-s + 0.903i·7-s + 0.316·9-s + 0.721·11-s + 0.667i·13-s + 0.0918i·15-s − 0.970·17-s + 1.09·19-s − 1.03i·21-s − 0.860i·23-s + 0.993·25-s + 0.783·27-s + 1.04i·29-s − 1.40i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7445705940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7445705940\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 30.9T + 729T^{2} \) |
| 5 | \( 1 + 10iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 309. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 960.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.46e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.76e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 7.52e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.04e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.54e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.18e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.99e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 2.93e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 2.15e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 7.56e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.92e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 7.84e4T + 4.21e10T^{2} \) |
| 61 | \( 1 + 1.09e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.94e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.88e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.10e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.04e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 3.10e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.45e6T + 8.32e11T^{2} \) |
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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
−11.48831948859150869802304705292, −10.61673173311100463773573408304, −9.305089225808063075806269845518, −8.655062966244507337781105152938, −7.05418056219885820227570753765, −6.23343770058290382328234963823, −5.33231755980442617460130377508, −4.31915632814042672422214760349, −2.63253947167963744735225251800, −1.12419677699335392670972983746,
0.26946140116293416852650998299, 1.28017627768555605728699843145, 3.22324946656302809175974682900, 4.52006658536806485683922633665, 5.51348799327458549648076510403, 6.58087770545728340164780112310, 7.34967332654867074688619999081, 8.697077066482396156140678144558, 9.913179823500062775042349076595, 10.76090153892993725244463287151
