L(s) = 1 | + 76.9i·3-s + 338. i·5-s + 438.·7-s − 3.73e3·9-s − 1.96e3i·11-s − 2.21e3i·13-s − 2.60e4·15-s − 1.21e4·17-s + 3.28e4i·19-s + 3.37e4i·21-s − 1.96e4·23-s − 3.64e4·25-s − 1.19e5i·27-s − 1.60e5i·29-s + 2.29e5·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s + 1.21i·5-s + 0.483·7-s − 1.70·9-s − 0.445i·11-s − 0.279i·13-s − 1.99·15-s − 0.598·17-s + 1.09i·19-s + 0.795i·21-s − 0.335·23-s − 0.466·25-s − 1.16i·27-s − 1.22i·29-s + 1.38·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.122934 + 1.45514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.122934 + 1.45514i\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 - 76.9iT - 2.18e3T^{2} \) |
| 5 | \( 1 - 338. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 438.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.96e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 2.21e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.21e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.28e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.60e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.96e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 5.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.83e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 8.20e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.53e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.82e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 4.84e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 7.98e4iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.53e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.89e4T + 8.07e13T^{2} \) |
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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
−15.59599693249637905423784086291, −14.86522233846568473604788621652, −13.90265782932642654623163269443, −11.58128283502675117952697091029, −10.64804278403974038528350927019, −9.801030355021572576741838527541, −8.169044837161255900095474211360, −6.09447505341967614494552090934, −4.39242488855139775717314783538, −2.98366391105781296884562579858,
0.69962857083990405606003934653, 2.00927200636710120099327563999, 4.91564704537888073003004333394, 6.65888563982908288301448129517, 7.962540343840501559336118560231, 9.049433442789260541264163078142, 11.36119240918799067039979003302, 12.49559070353605469816910561760, 13.15386798910783165390003917362, 14.33209996447555037023327097422