| L(s) = 1 | − 48i·5-s − 81·9-s + 240i·13-s − 322·17-s − 1.67e3·25-s − 1.68e3i·29-s + 1.68e3i·37-s − 3.03e3·41-s + 3.88e3i·45-s + 2.40e3·49-s − 5.04e3i·53-s − 2.64e3i·61-s + 1.15e4·65-s − 1.44e3·73-s + 6.56e3·81-s + ⋯ |
| L(s) = 1 | − 1.91i·5-s − 9-s + 1.42i·13-s − 1.11·17-s − 2.68·25-s − 1.99i·29-s + 1.22i·37-s − 1.80·41-s + 1.92i·45-s + 49-s − 1.79i·53-s − 0.709i·61-s + 2.72·65-s − 0.270·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(-0.594394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.594394i\) |
| \(L(3)\) |
|
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 | \( 1 + 81T^{2} \) |
| 5 | \( 1 + 48iT - 625T^{2} \) |
| 7 | \( 1 - 2.40e3T^{2} \) |
| 11 | \( 1 + 1.46e4T^{2} \) |
| 13 | \( 1 - 240iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 322T + 8.35e4T^{2} \) |
| 19 | \( 1 + 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.68e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.68e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 3.03e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 + 5.04e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.64e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.44e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 + 4.74e7T^{2} \) |
| 89 | \( 1 - 9.75e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.91e3T + 8.85e7T^{2} \) |
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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
−11.95476100072361325686411129592, −11.52611765627791324528078100774, −9.707971330735629480825278745742, −8.836027053651384999741107747022, −8.203499844021390198239163590562, −6.41547940000090852842055724284, −5.12748302862244653429947865523, −4.17459121328282793653302917306, −1.92785120546766015276106085234, −0.23211058303457400055487928195,
2.54786609751923929910705607549, 3.44484860476081351579987970721, 5.53828172783118005202655226370, 6.62643202183532143575609325941, 7.61129217988447222760787850012, 8.908278347486594212332521711345, 10.52781780887791437533400036520, 10.77856927981860004257973790983, 11.92268140272457656008882950569, 13.33091461284625245745419418150
