L(s) = 1 | + (−1.71 − 0.988i)2-s + (1.03 + 1.38i)3-s + (0.952 + 1.65i)4-s + (−0.866 + 0.5i)5-s + (−0.410 − 3.39i)6-s + (−1.22 − 0.707i)7-s + 0.186i·8-s + (−0.837 + 2.88i)9-s + 1.97·10-s + (−2.00 − 1.15i)11-s + (−1.29 + 3.03i)12-s + (−0.495 − 3.57i)13-s + (1.39 + 2.42i)14-s + (−1.59 − 0.679i)15-s + (2.08 − 3.61i)16-s − 3.18·17-s + ⋯ |
L(s) = 1 | + (−1.21 − 0.698i)2-s + (0.600 + 0.799i)3-s + (0.476 + 0.825i)4-s + (−0.387 + 0.223i)5-s + (−0.167 − 1.38i)6-s + (−0.463 − 0.267i)7-s + 0.0658i·8-s + (−0.279 + 0.960i)9-s + 0.624·10-s + (−0.604 − 0.348i)11-s + (−0.373 + 0.876i)12-s + (−0.137 − 0.990i)13-s + (0.373 + 0.647i)14-s + (−0.411 − 0.175i)15-s + (0.522 − 0.904i)16-s − 0.772·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.114517 - 0.300773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.114517 - 0.300773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.03 - 1.38i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.495 + 3.57i)T \) |
good | 2 | \( 1 + (1.71 + 0.988i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (1.22 + 0.707i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.00 + 1.15i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 0.980iT - 19T^{2} \) |
| 23 | \( 1 + (1.24 + 2.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.20 + 3.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.28iT - 37T^{2} \) |
| 41 | \( 1 + (-2.06 + 1.19i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.60 + 7.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.30 + 3.06i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.53T + 53T^{2} \) |
| 59 | \( 1 + (-1.50 + 0.871i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.11 - 3.65i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.5 - 6.69i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.496iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + (2.20 - 3.81i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.14 + 3.55i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9.63iT - 89T^{2} \) |
| 97 | \( 1 + (10.4 + 6.06i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.26190309771539130213113376572, −9.692714944279581073702614766744, −8.655411980455620515212176491962, −8.164147994522845107971688278451, −7.25888958534949568113180422118, −5.69556652833754012085156559875, −4.44185496157277760314985714166, −3.17500329914258083960811960847, −2.40273567450664355481963900213, −0.24734709009938473014647997992,
1.53679041738294416528333855185, 3.02619278336499111159816906206, 4.48789652233740178468300345700, 6.21087563239708844832784138129, 6.77854292088672878928955238973, 7.72261029299673077992105449753, 8.245533634761428165134838105507, 9.216439729129217527033058239754, 9.585670998378455638381941918920, 10.82993615117613688341791290467