L(s) = 1 | + (0.0150 + 0.0261i)3-s − 0.691i·5-s + (−0.866 − 0.5i)7-s + (1.49 − 2.59i)9-s + (4.62 − 2.66i)11-s + (−3.55 + 0.629i)13-s + (0.0180 − 0.0104i)15-s + (−2.67 + 4.63i)17-s + (−3.86 − 2.23i)19-s − 0.0301i·21-s + (−2.95 − 5.12i)23-s + 4.52·25-s + 0.181·27-s + (−0.0499 − 0.0865i)29-s − 10.4i·31-s + ⋯ |
L(s) = 1 | + (0.00871 + 0.0150i)3-s − 0.309i·5-s + (−0.327 − 0.188i)7-s + (0.499 − 0.865i)9-s + (1.39 − 0.804i)11-s + (−0.984 + 0.174i)13-s + (0.00466 − 0.00269i)15-s + (−0.648 + 1.12i)17-s + (−0.887 − 0.512i)19-s − 0.00658i·21-s + (−0.617 − 1.06i)23-s + 0.904·25-s + 0.0348·27-s + (−0.00927 − 0.0160i)29-s − 1.87i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03596 - 0.879888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03596 - 0.879888i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (3.55 - 0.629i)T \) |
good | 3 | \( 1 + (-0.0150 - 0.0261i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 0.691iT - 5T^{2} \) |
| 11 | \( 1 + (-4.62 + 2.66i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.67 - 4.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.86 + 2.23i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.95 + 5.12i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0499 + 0.0865i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (-5.37 + 3.10i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.50 + 4.33i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.42 + 5.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.94iT - 47T^{2} \) |
| 53 | \( 1 + 5.49T + 53T^{2} \) |
| 59 | \( 1 + (-6.31 - 3.64i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.0818 + 0.141i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.2 - 6.48i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.35 + 2.51i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.8iT - 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 + (-1.68 + 0.970i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.8 - 9.15i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.15759066604450764125446000841, −9.138560459458672603070811391983, −8.824117956269984626200684488894, −7.53405528331494686749813918937, −6.47227671403744190223280503387, −6.08649159016698927613905232333, −4.34121868507540704387108958861, −3.96591643718808607247082036698, −2.36391025887752896744804329131, −0.72337146603765723112317076703,
1.71139241255683237059950520803, 2.88454786563821146866608576597, 4.29133648805382855880611513825, 4.99754119145837979041974050560, 6.39137424896062799870881894051, 7.05768195209904372374148876401, 7.84218737041494844414111578971, 9.091676624535626007923134468334, 9.692071756719286720343320815124, 10.49825532610852868091742621449