L(s) = 1 | − 21.6i·3-s + 25i·5-s − 169.·7-s − 224.·9-s + 512. i·11-s + 36.4i·13-s + 540.·15-s − 1.26e3·17-s − 790. i·19-s + 3.66e3i·21-s + 4.99e3·23-s − 625·25-s − 401. i·27-s − 934. i·29-s + 6.69e3·31-s + ⋯ |
L(s) = 1 | − 1.38i·3-s + 0.447i·5-s − 1.30·7-s − 0.923·9-s + 1.27i·11-s + 0.0598i·13-s + 0.620·15-s − 1.05·17-s − 0.502i·19-s + 1.81i·21-s + 1.96·23-s − 0.200·25-s − 0.106i·27-s − 0.206i·29-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.483656199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483656199\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
good | 3 | \( 1 + 21.6iT - 243T^{2} \) |
| 7 | \( 1 + 169.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 512. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 36.4iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 790. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.99e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 934. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.30e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.37e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.32e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.77e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.54e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 435.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.94e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 6.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.59e5T + 8.58e9T^{2} \) |
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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
−10.72513223677904618634824139695, −9.698016547477090520834433638836, −8.780942419417228612815915490930, −7.38782648165554503754171025618, −6.86827289382220757817772447900, −6.27873656536553628737719409244, −4.66000573876479079074735998652, −3.03612911067294077104910799486, −2.11761132718205947878400243489, −0.67255185256065740448129217414,
0.65095184621919858008266200682, 2.91860886808259766676361948715, 3.71738392175713744946768507023, 4.81440405908055969236320330557, 5.83929869812138479403060197499, 6.87497974501507718872713462218, 8.576974510600062909538269585384, 9.082527076286042027554335479488, 9.968588823736185335237431841013, 10.73815998735997763507907612163