| L(s) = 1 | + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−1.18 − 1.36i)5-s + (0.959 − 0.281i)6-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−1.52 + 0.978i)10-s + (0.294 + 2.04i)11-s + (−0.142 − 0.989i)12-s + (4.53 − 2.91i)13-s + (0.654 − 0.755i)14-s + (0.751 − 1.64i)15-s + (0.841 + 0.540i)16-s + (0.123 − 0.0362i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.529 − 0.611i)5-s + (0.391 − 0.115i)6-s + (0.317 + 0.204i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.481 + 0.309i)10-s + (0.0886 + 0.616i)11-s + (−0.0410 − 0.285i)12-s + (1.25 − 0.808i)13-s + (0.175 − 0.201i)14-s + (0.194 − 0.424i)15-s + (0.210 + 0.135i)16-s + (0.0299 − 0.00878i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52278 - 0.739445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52278 - 0.739445i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (3.43 - 3.34i)T \) |
| good | 5 | \( 1 + (1.18 + 1.36i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.294 - 2.04i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-4.53 + 2.91i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.123 + 0.0362i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.21 - 0.651i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-9.94 + 2.92i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.42 + 5.31i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.33 + 5.00i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.31 - 4.97i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.36 + 7.36i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 + (-3.47 - 2.23i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (4.12 - 2.65i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.97 + 8.69i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.21 - 8.47i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.419 + 2.91i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.4 - 3.36i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.66 - 1.07i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.11 + 4.75i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.19 - 11.3i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (0.372 + 0.429i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−9.958865214251095388360289345039, −9.182635906032813080679298016882, −8.229739158318106230248572994451, −7.893525443947122548085620244734, −6.24904030926403684995014304478, −5.27692356021167374472181581996, −4.36469626137064819794452585333, −3.67811159035644876767244421080, −2.46530427714375978765652325039, −0.971397201084020024146937508971,
1.17789693740466780931095099389, 2.95057513912190237681510388915, 3.85166305004024237937667336285, 4.91161426186251260178057126670, 6.28390497711077325111334294484, 6.61473152286620100072638273853, 7.61776672217938981835150465789, 8.361183997210511583642049276360, 8.897283721606246673430367517209, 10.13033655419549545309421413625
