L(s) = 1 | + (−0.962 + 1.44i)3-s + (2.75 + 2.31i)5-s + (−1.28 − 0.467i)7-s + (−1.14 − 2.77i)9-s + (−0.884 + 0.742i)11-s + (−1.09 + 6.20i)13-s + (−5.97 + 1.74i)15-s + (0.526 − 0.911i)17-s + (1.05 + 1.82i)19-s + (1.91 − 1.40i)21-s + (6.16 − 2.24i)23-s + (1.37 + 7.80i)25-s + (5.09 + 1.01i)27-s + (−1.37 − 7.80i)29-s + (−5.95 + 2.16i)31-s + ⋯ |
L(s) = 1 | + (−0.555 + 0.831i)3-s + (1.23 + 1.03i)5-s + (−0.485 − 0.176i)7-s + (−0.382 − 0.923i)9-s + (−0.266 + 0.223i)11-s + (−0.303 + 1.72i)13-s + (−1.54 + 0.449i)15-s + (0.127 − 0.221i)17-s + (0.241 + 0.418i)19-s + (0.416 − 0.305i)21-s + (1.28 − 0.467i)23-s + (0.275 + 1.56i)25-s + (0.980 + 0.195i)27-s + (−0.255 − 1.44i)29-s + (−1.07 + 0.389i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0364 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0364 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.772430 + 0.801088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.772430 + 0.801088i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.962 - 1.44i)T \) |
good | 5 | \( 1 + (-2.75 - 2.31i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.28 + 0.467i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.884 - 0.742i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.09 - 6.20i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.526 + 0.911i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.05 - 1.82i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.16 + 2.24i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.37 + 7.80i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.95 - 2.16i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.41 + 7.64i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.448 + 2.54i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.115 + 0.0967i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.75 - 3.54i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 2.09T + 53T^{2} \) |
| 59 | \( 1 + (-3.16 - 2.65i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-8.44 - 3.07i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.79 + 10.1i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-8.05 + 13.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.63 - 2.83i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.318 - 1.80i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.68 - 9.55i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.14 + 5.44i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.4 - 8.73i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.48420857935401268233021215738, −11.29645740683526695936571451369, −10.60803764887225751315645481492, −9.623862375075537588521186185213, −9.245956985271275813301429950183, −7.12288957086148455608824761629, −6.37305887730617714776576153954, −5.35893607211014954084973223854, −3.94059620867908063599657070948, −2.42499598844662993933408890422,
1.08968177680607266662570154092, 2.77268089385271338762103812410, 5.31516915635850860128312552693, 5.50786646293906274416944986969, 6.85085815897601543473350794590, 8.080629892369759971117850695927, 9.119866610652866862642181544794, 10.13449092675887433173837632816, 11.12119911325032464981975786686, 12.49735545997619225976385236881