L(s) = 1 | + (1.29 + 2.23i)3-s + (−0.5 + 0.866i)5-s + (0.292 + 2.62i)7-s + (−1.83 + 3.18i)9-s + (0.839 + 1.45i)11-s − 4.84·13-s − 2.58·15-s + (−1 − 1.73i)17-s + (3.42 − 5.92i)19-s + (−5.50 + 4.05i)21-s + (−1.13 + 1.95i)23-s + (−0.499 − 0.866i)25-s − 1.75·27-s + 3.32·29-s + (4.58 + 7.94i)31-s + ⋯ |
L(s) = 1 | + (0.746 + 1.29i)3-s + (−0.223 + 0.387i)5-s + (0.110 + 0.993i)7-s + (−0.613 + 1.06i)9-s + (0.253 + 0.438i)11-s − 1.34·13-s − 0.667·15-s + (−0.242 − 0.420i)17-s + (0.785 − 1.36i)19-s + (−1.20 + 0.884i)21-s + (−0.235 + 0.408i)23-s + (−0.0999 − 0.173i)25-s − 0.337·27-s + 0.616·29-s + (0.823 + 1.42i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.692208 + 1.48343i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692208 + 1.48343i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.292 - 2.62i)T \) |
good | 3 | \( 1 + (-1.29 - 2.23i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.839 - 1.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.42 + 5.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.13 - 1.95i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + (-4.58 - 7.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.42 + 2.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 + (6.10 - 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.74 + 6.48i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 + 5.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.87 + 4.98i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.84 + 4.93i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + (-2.92 + 5.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.99971057892255777996606684391, −9.860637163999855245043944754611, −9.488180177103191866153964675733, −8.696811912496493460013725820250, −7.66672781770612610469313240008, −6.62545112586096912593206479359, −5.07486358305256984886810395388, −4.61985150353475794569801361096, −3.17490205068513814529315321161, −2.49401664589686469415010324551,
0.884312786195528102555246972219, 2.17477049737854135177959879139, 3.50899894985610989971206169336, 4.67296260121069726689518210064, 6.13386418614711694049940757199, 7.06071490457374681036665045941, 7.88585366193570401120563335419, 8.237310606637403171778303255239, 9.531203281717350902280027868699, 10.31328483446710503081977976853