| L(s) = 1 | + (0.263 + 0.184i)2-s + (−0.312 + 1.70i)3-s + (−0.648 − 1.78i)4-s + (2.21 − 0.303i)5-s + (−0.396 + 0.390i)6-s + (2.07 + 4.46i)7-s + (0.324 − 1.20i)8-s + (−2.80 − 1.06i)9-s + (0.639 + 0.328i)10-s + (0.0393 + 0.0469i)11-s + (3.23 − 0.548i)12-s + (0.0455 + 0.0650i)13-s + (−0.274 + 1.55i)14-s + (−0.174 + 3.86i)15-s + (−2.59 + 2.17i)16-s + (−1.04 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (0.186 + 0.130i)2-s + (−0.180 + 0.983i)3-s + (−0.324 − 0.891i)4-s + (0.990 − 0.135i)5-s + (−0.161 + 0.159i)6-s + (0.786 + 1.68i)7-s + (0.114 − 0.427i)8-s + (−0.934 − 0.354i)9-s + (0.202 + 0.103i)10-s + (0.0118 + 0.0141i)11-s + (0.935 − 0.158i)12-s + (0.0126 + 0.0180i)13-s + (−0.0734 + 0.416i)14-s + (−0.0449 + 0.998i)15-s + (−0.649 + 0.544i)16-s + (−0.252 − 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15167 + 0.438602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15167 + 0.438602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.312 - 1.70i)T \) |
| 5 | \( 1 + (-2.21 + 0.303i)T \) |
| good | 2 | \( 1 + (-0.263 - 0.184i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-2.07 - 4.46i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.0393 - 0.0469i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0455 - 0.0650i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.04 + 3.89i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.80 + 1.04i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.88 + 2.74i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (1.24 + 7.06i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.209 + 0.0761i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (6.33 - 1.69i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.22 + 0.391i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.325 - 3.71i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.26 + 0.591i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-8.67 - 8.67i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.888 + 0.745i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.77 + 2.46i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.22 + 4.36i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-6.33 - 3.65i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.42 - 1.72i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.06 - 1.06i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.31 - 3.30i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (5.25 + 9.09i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.77 - 0.855i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.75406210257366708228056634349, −12.18599808402860416740432811368, −11.24005668565835470713349231215, −10.04661270011829270913192311126, −9.338934197658601396437595655298, −8.558800742818945027256331061624, −6.18252662264153677667586824175, −5.46374420240944417242663754706, −4.68430272236344319913969143532, −2.32686251463718679070234882435,
1.74764573623712245821144162388, 3.70778026949637823132341505698, 5.26487786212937431719109952917, 6.80532607512202989795740077361, 7.65494804990285269459171928192, 8.627845887487710553575461360649, 10.26159657211250872236789615506, 11.16735229475652656748904242015, 12.31586017744825178674866338853, 13.27986198678944976477621129269
