| L(s) = 1 | − 153.·5-s + 395. i·7-s − 18.9i·11-s − 4.24e3·13-s + 987.·17-s + 9.17e3i·19-s + 1.76e4i·23-s + 8.06e3·25-s − 7.62e3·29-s + 1.73e4i·31-s − 6.09e4i·35-s − 6.51e4·37-s − 9.26e4·41-s + 8.14e4i·43-s + 3.07e4i·47-s + ⋯ |
| L(s) = 1 | − 1.23·5-s + 1.15i·7-s − 0.0142i·11-s − 1.93·13-s + 0.201·17-s + 1.33i·19-s + 1.45i·23-s + 0.516·25-s − 0.312·29-s + 0.581i·31-s − 1.42i·35-s − 1.28·37-s − 1.34·41-s + 1.02i·43-s + 0.295i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2466574475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2466574475\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 153.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 395. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 18.9iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 4.24e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 987.T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.17e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.76e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 7.62e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.73e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 6.51e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 9.26e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 8.14e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 3.07e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.51e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.62e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.39e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 9.48e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 2.98e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.78e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.68e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.66e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.04e6T + 4.96e11T^{2} \) |
| 97 | \( 1 + 4.01e5T + 8.32e11T^{2} \) |
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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
−10.33352566430296271836153971182, −9.533839189894390878673207323466, −8.577694381604355067013396050518, −7.73304055842220759561227891152, −7.12477637321691325691517318491, −5.71439137501411822882429561337, −4.98014030369910034076402066987, −3.80307318486479069451312939098, −2.84922035080885109918587966218, −1.65294648423027832004302631771,
0.093029998859296159009182677036, 0.51326741321247126298755596064, 2.27784788517244234332258463142, 3.47014509481772347683306839133, 4.41693432082728609318147635324, 5.05887288836330411075623019501, 6.84632471096722999995486810654, 7.24104188662609699896289618102, 8.036127844266321068674865613328, 9.066152265656781484860977390890
