L(s) = 1 | + (−1.20 + 2.08i)2-s + (1.64 − 0.539i)3-s + (−1.90 − 3.29i)4-s + (−0.5 − 0.866i)5-s + (−0.857 + 4.08i)6-s + (−0.497 + 0.861i)7-s + 4.36·8-s + (2.41 − 1.77i)9-s + 2.41·10-s + (−2.77 + 4.80i)11-s + (−4.91 − 4.40i)12-s + (0.5 + 0.866i)13-s + (−1.19 − 2.07i)14-s + (−1.29 − 1.15i)15-s + (−1.44 + 2.50i)16-s + 5.33·17-s + ⋯ |
L(s) = 1 | + (−0.852 + 1.47i)2-s + (0.950 − 0.311i)3-s + (−0.952 − 1.64i)4-s + (−0.223 − 0.387i)5-s + (−0.350 + 1.66i)6-s + (−0.188 + 0.325i)7-s + 1.54·8-s + (0.806 − 0.591i)9-s + 0.762·10-s + (−0.837 + 1.44i)11-s + (−1.41 − 1.27i)12-s + (0.138 + 0.240i)13-s + (−0.320 − 0.555i)14-s + (−0.333 − 0.298i)15-s + (−0.361 + 0.626i)16-s + 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.601382 + 0.967843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.601382 + 0.967843i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.64 + 0.539i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (1.20 - 2.08i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (0.497 - 0.861i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.77 - 4.80i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 4.89i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.44 - 5.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.67 - 2.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 + (-5.36 - 9.30i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.80 + 10.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.320 - 0.554i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.0369T + 53T^{2} \) |
| 59 | \( 1 + (6.89 + 11.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.48 - 2.56i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.83 + 8.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 + (0.938 - 1.62i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.80 + 10.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + (2.93 - 5.08i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.42762123812127114577661496295, −9.450214816004397731327262608690, −9.252997929668348877653067577633, −8.081636609094221210895805524576, −7.51490475249644879344557340469, −7.02048547590028551340586503043, −5.66312851948741998936056371842, −4.81909239099172280689786674376, −3.20077611260353481770955634933, −1.43285430644078248323160212969,
0.872485513590632478588703694950, 2.62671687019353100053525633168, 3.18591074653952959789123609265, 4.06006049428782341655966030529, 5.69561537318163879969779946316, 7.50393739080898766325451460826, 8.044266724107510976079799088476, 8.874496883153990178111975207377, 9.709934156626213240832186757947, 10.45115300717786148054158542041