| L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s + (0.279 − 2.66i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (−5.62 + 2.50i)11-s + (0.978 − 0.207i)12-s + (−2.81 − 0.598i)13-s + (−2.44 − 1.08i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (2.86 + 1.27i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.386 + 0.429i)3-s + (−0.404 − 0.293i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (0.105 − 1.00i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (−1.69 + 0.755i)11-s + (0.282 − 0.0600i)12-s + (−0.780 − 0.165i)13-s + (−0.653 − 0.290i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.694 + 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0574492 + 0.154249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0574492 + 0.154249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.356 - 5.55i)T \) |
| good | 7 | \( 1 + (-0.279 + 2.66i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (5.62 - 2.50i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (2.81 + 0.598i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.86 - 1.27i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.54 - 0.328i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (1.58 - 1.14i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.815 - 2.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (5.42 + 9.39i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.02 + 7.80i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (6.21 - 1.32i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.86 - 8.81i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.849 - 8.08i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-5.49 + 6.09i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + (3.47 - 6.02i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.739 + 7.03i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 0.720i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (15.4 + 6.86i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.231 - 0.256i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-6.84 - 4.97i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-11.0 - 8.05i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.968384893110829227631843651497, −8.955523582856643354904928135303, −7.81574811500642795713108219570, −7.13366304142324193135757186992, −5.64712506602802854560318242731, −5.03251967626744981058262931275, −4.24319446413991671362278572151, −3.09819808930790112772603759538, −1.76105633305243695212699564963, −0.07038077519518455138621670679,
2.24964910288865027582020878688, 3.14973708180989475823857475922, 4.88119019086926934499154525495, 5.46956147239066263333491625204, 6.18433964631304454744427837243, 7.14658016245236880756739494981, 8.027495272538550619236940150879, 8.557022999604785531739804161008, 9.836758547381335944486773773756, 10.43236364696382901782476789788
