L(s) = 1 | + (−3.36 + 4.55i)2-s + (−18.2 + 18.2i)3-s + (−9.40 − 30.5i)4-s + (−17.6 − 17.6i)5-s + (−21.7 − 144. i)6-s + 13.9i·7-s + (170. + 60.0i)8-s − 424. i·9-s + (139. − 21.0i)10-s + (−220. − 220. i)11-s + (730. + 386. i)12-s + (433. − 433. i)13-s + (−63.3 − 46.7i)14-s + 645.·15-s + (−847. + 575. i)16-s − 25.9·17-s + ⋯ |
L(s) = 1 | + (−0.594 + 0.804i)2-s + (−1.17 + 1.17i)3-s + (−0.293 − 0.955i)4-s + (−0.316 − 0.316i)5-s + (−0.246 − 1.63i)6-s + 0.107i·7-s + (0.943 + 0.331i)8-s − 1.74i·9-s + (0.442 − 0.0664i)10-s + (−0.549 − 0.549i)11-s + (1.46 + 0.775i)12-s + (0.711 − 0.711i)13-s + (−0.0863 − 0.0637i)14-s + 0.741·15-s + (−0.827 + 0.561i)16-s − 0.0217·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.526879 + 0.284203i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.526879 + 0.284203i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.36 - 4.55i)T \) |
| 5 | \( 1 + (17.6 + 17.6i)T \) |
good | 3 | \( 1 + (18.2 - 18.2i)T - 243iT^{2} \) |
| 7 | \( 1 - 13.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (220. + 220. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (-433. + 433. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + 25.9T + 1.41e6T^{2} \) |
| 19 | \( 1 + (1.56e3 - 1.56e3i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 1.49e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-324. + 324. i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 - 7.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (3.03e3 + 3.03e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.75e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.12e4 - 1.12e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 - 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.53e4 - 1.53e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (2.44e4 + 2.44e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (2.89e4 - 2.89e4i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (-4.27e4 + 4.27e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 4.05e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 872. iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (6.30e4 - 6.30e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 5.13e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.17e5T + 8.58e9T^{2} \) |
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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
−13.82505266607450294389251549769, −12.26377956934416533746892023183, −10.85867930421860311020262366608, −10.41938119015863022523559257516, −9.102263391869209817387843952494, −7.964639255917187149443985475914, −6.14486829371796277897052291805, −5.41211844624521882078388757527, −4.10456122588173588781062531291, −0.57693108845442145728265938597,
0.790270007198244723198669428583, 2.32042568726959130899947111211, 4.52827156193810192224032692694, 6.45585612310664059862558975102, 7.36493183979755074646786115174, 8.606800709968852730029763892957, 10.35507301282979909303109747439, 11.17768743558050979008162649533, 11.99349597513179494588191775914, 12.86807943076369521916540803385