| L(s) = 1 | − 51.1i·3-s − 142.·5-s + 217. i·7-s − 1.89e3·9-s − 25.7i·11-s + 1.39e3·13-s + 7.30e3i·15-s + 4.00e3·17-s + 129. i·19-s + 1.11e4·21-s − 3.69e3i·23-s + 4.75e3·25-s + 5.95e4i·27-s − 4.08e4·29-s + 4.61e4i·31-s + ⋯ |
| L(s) = 1 | − 1.89i·3-s − 1.14·5-s + 0.634i·7-s − 2.59·9-s − 0.0193i·11-s + 0.632·13-s + 2.16i·15-s + 0.814·17-s + 0.0188i·19-s + 1.20·21-s − 0.303i·23-s + 0.304·25-s + 3.02i·27-s − 1.67·29-s + 1.54i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0350729 + 0.0350729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0350729 + 0.0350729i\) |
| \(L(4)\) |
|
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 | \( 1 + 51.1iT - 729T^{2} \) |
| 5 | \( 1 + 142.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 217. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 25.7iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.39e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.00e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 129. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.69e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 4.08e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 6.59e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 4.18e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 3.00e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.97e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.37e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.43e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.06e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.49e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.66e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.52e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.97e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 8.78e3T + 8.32e11T^{2} \) |
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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
−13.78730956117695594182142570618, −12.60746012710607549175111652826, −12.03943103459112158398929499318, −11.10269860109307692124386935518, −8.761367180851413911492194306100, −7.907254453125347712791374284697, −6.95158013481528262387135852219, −5.64545404479473427358053032818, −3.23971247456869825042092128095, −1.53375598296856152202324958330,
0.02046601963273003907447040975, 3.51021301500185844490881487550, 4.12207212493215033941193549040, 5.54805874615742604307455263215, 7.67709223800213472962165448416, 8.908229020382438024900186501019, 10.07905115145154686531599612525, 11.00877463588594232012488861871, 11.82383677985741882014529725028, 13.71069904144195160684871027576
