| L(s) = 1 | + (0.819 + 0.573i)2-s + (−0.905 + 1.47i)3-s + (0.342 + 0.939i)4-s + (2.22 − 0.218i)5-s + (−1.58 + 0.690i)6-s + (−3.72 + 0.997i)7-s + (−0.258 + 0.965i)8-s + (−1.36 − 2.67i)9-s + (1.94 + 1.09i)10-s + (−5.37 + 3.10i)11-s + (−1.69 − 0.345i)12-s + (0.170 − 0.0148i)13-s + (−3.62 − 1.31i)14-s + (−1.69 + 3.48i)15-s + (−0.766 + 0.642i)16-s + (1.47 + 1.03i)17-s + ⋯ |
| L(s) = 1 | + (0.579 + 0.405i)2-s + (−0.522 + 0.852i)3-s + (0.171 + 0.469i)4-s + (0.995 − 0.0975i)5-s + (−0.648 + 0.281i)6-s + (−1.40 + 0.377i)7-s + (−0.0915 + 0.341i)8-s + (−0.453 − 0.891i)9-s + (0.616 + 0.347i)10-s + (−1.62 + 0.936i)11-s + (−0.489 − 0.0997i)12-s + (0.0471 − 0.00412i)13-s + (−0.968 − 0.352i)14-s + (−0.436 + 0.899i)15-s + (−0.191 + 0.160i)16-s + (0.358 + 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0870227 + 1.15991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0870227 + 1.15991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.819 - 0.573i)T \) |
| 3 | \( 1 + (0.905 - 1.47i)T \) |
| 5 | \( 1 + (-2.22 + 0.218i)T \) |
| 19 | \( 1 + (4.29 - 0.734i)T \) |
| good | 7 | \( 1 + (3.72 - 0.997i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (5.37 - 3.10i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.170 + 0.0148i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.47 - 1.03i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-6.69 - 3.12i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (1.14 - 6.51i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.994 - 1.72i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.60 + 5.60i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.31 - 2.75i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.86 + 2.27i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.87 + 2.67i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-5.69 - 2.65i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (0.280 + 1.58i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.41 - 2.33i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-7.00 + 4.90i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-1.55 + 4.26i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.30 - 14.9i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-1.71 - 2.04i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.8 + 3.17i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.82 - 4.04i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (8.06 - 11.5i)T + (-33.1 - 91.1i)T^{2} \) |
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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
−10.83130292629790321090989394652, −10.34819380790242364379288612814, −9.480223815680463634041501377462, −8.795550810005256449395349148630, −7.23303710937013036984287879327, −6.37748368249271720958951422397, −5.48450400727608201284850715607, −4.99325480309208438499525108938, −3.53465313400966602985681462304, −2.52456614672317889673365129162,
0.54011960395716449979237100950, 2.38750354503424976242627163867, 3.12040315038320568537932438036, 4.92255217277667065732173907434, 5.85174183962472947327391724689, 6.38873735817957218011688929019, 7.31354918327358567077502274684, 8.578920841717670160868385953988, 9.759403988320741009731129629656, 10.58976705898568926468693525855
