| L(s) = 1 | − 20.7i·3-s + 35.2i·7-s − 186.·9-s + 7.33·11-s + 619. i·13-s − 959. i·17-s − 309.·19-s + 731.·21-s − 2.46e3i·23-s − 1.16e3i·27-s + 1.28e3·29-s − 7.09e3·31-s − 152. i·33-s + 6.10e3i·37-s + 1.28e4·39-s + ⋯ |
| L(s) = 1 | − 1.33i·3-s + 0.272i·7-s − 0.768·9-s + 0.0182·11-s + 1.01i·13-s − 0.805i·17-s − 0.196·19-s + 0.361·21-s − 0.972i·23-s − 0.307i·27-s + 0.283·29-s − 1.32·31-s − 0.0242i·33-s + 0.732i·37-s + 1.35·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.324930657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324930657\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 20.7iT - 243T^{2} \) |
| 7 | \( 1 - 35.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 7.33T + 1.61e5T^{2} \) |
| 13 | \( 1 - 619. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 959. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 309.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.46e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.10e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.14e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.37e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.54e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.55e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.86e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.11e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.76e5iT - 8.58e9T^{2} \) |
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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
−9.374711717098917181296967713131, −8.642213747427049518719009191835, −7.74044260314471811366856268631, −6.90830134088066661176895212632, −6.41416707952141370898844221893, −5.28070950519493830917936157897, −4.17011942592740258424259674521, −2.74814339955916612785425871837, −1.90288153192128656929405176567, −0.914073596376324113631924768096,
0.30573227104282507814001161180, 1.84676435917851977085490339488, 3.39707517875015422181709895450, 3.82734816909874302145697322284, 5.02939045619575931173859515439, 5.60223697241290148346940235470, 6.84715233768553034674696652107, 7.899989233497910475756549600934, 8.752150371613434608555329701640, 9.598645451896618701476418953362
