| L(s) = 1 | + (−0.327 + 0.945i)2-s + (0.458 + 0.888i)3-s + (−0.786 − 0.618i)4-s + (1.26 − 0.120i)5-s + (−0.989 + 0.142i)6-s + (−2.41 − 1.08i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.298 + 1.23i)10-s + (−4.57 + 1.58i)11-s + (0.189 − 0.981i)12-s + (0.667 + 2.27i)13-s + (1.81 − 1.92i)14-s + (0.685 + 1.06i)15-s + (0.235 + 0.971i)16-s + (−3.75 − 1.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (0.564 − 0.0538i)5-s + (−0.404 + 0.0580i)6-s + (−0.911 − 0.411i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0944 + 0.389i)10-s + (−1.37 + 0.476i)11-s + (0.0546 − 0.283i)12-s + (0.185 + 0.630i)13-s + (0.485 − 0.513i)14-s + (0.176 + 0.275i)15-s + (0.0589 + 0.242i)16-s + (−0.911 − 0.364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0869449 - 0.198945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0869449 - 0.198945i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.327 - 0.945i)T \) |
| 3 | \( 1 + (-0.458 - 0.888i)T \) |
| 7 | \( 1 + (2.41 + 1.08i)T \) |
| 23 | \( 1 + (-0.901 + 4.71i)T \) |
| good | 5 | \( 1 + (-1.26 + 0.120i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (4.57 - 1.58i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-0.667 - 2.27i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (3.75 + 1.50i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-1.21 + 0.486i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.617 - 4.29i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (7.09 - 0.337i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-1.40 - 1.00i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (10.4 + 4.77i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (0.853 - 1.32i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (7.44 - 4.29i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.65 - 2.78i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (4.11 + 0.997i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-7.07 - 3.64i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (1.75 + 9.11i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (7.47 + 8.62i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-8.18 + 10.4i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-1.70 + 1.78i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-5.60 - 12.2i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.436 - 9.15i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (-3.46 + 7.59i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.39701110456071888028837091489, −9.521127270860766952891598403942, −9.079321846855223814854125844371, −8.044776405470218419925928928485, −7.10272181078967234006566307421, −6.41981476214110054443772464444, −5.32670851546268531962418621561, −4.59421251805925399081364364515, −3.37039933501114762941298911508, −2.12844257822314334077254797746,
0.095525965923975626470734480066, 1.90492908835669047097227222157, 2.81542462786922331570037669137, 3.64626828425957178832324283846, 5.29780885030734104695556295845, 5.93812010062491844031334289707, 7.02508203891133863444778729559, 8.048052716331169658738891110721, 8.672591527542507491950838232593, 9.687945410722179763469348080950
