L(s) = 1 | − 2.77·5-s + (0.855 + 2.50i)7-s + 3.43·11-s + (−0.429 + 0.743i)13-s + (0.405 − 0.701i)17-s + (0.750 + 1.29i)19-s − 7.64·23-s + 2.68·25-s + (−3.99 − 6.92i)29-s + (3.60 + 6.24i)31-s + (−2.37 − 6.93i)35-s + (0.458 + 0.793i)37-s + (−1.67 + 2.90i)41-s + (1.20 + 2.08i)43-s + (−0.307 + 0.532i)47-s + ⋯ |
L(s) = 1 | − 1.23·5-s + (0.323 + 0.946i)7-s + 1.03·11-s + (−0.119 + 0.206i)13-s + (0.0982 − 0.170i)17-s + (0.172 + 0.298i)19-s − 1.59·23-s + 0.536·25-s + (−0.742 − 1.28i)29-s + (0.647 + 1.12i)31-s + (−0.400 − 1.17i)35-s + (0.0753 + 0.130i)37-s + (−0.261 + 0.453i)41-s + (0.183 + 0.318i)43-s + (−0.0448 + 0.0776i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6943738589\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6943738589\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.855 - 2.50i)T \) |
good | 5 | \( 1 + 2.77T + 5T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + (0.429 - 0.743i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.405 + 0.701i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.750 - 1.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 + (3.99 + 6.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.60 - 6.24i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.458 - 0.793i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.67 - 2.90i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.20 - 2.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.307 - 0.532i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.31 - 10.9i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.734 - 1.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.71 - 9.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.10 + 14.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + (4.16 - 7.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.75 - 9.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.11 - 8.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.605554497318809202694610508153, −8.957948767904428876886101066383, −8.060021332826828852161319790214, −7.64846425875397311124208649032, −6.48383511464737133205967892496, −5.78444834284872365121631997384, −4.56005137186555250014317091801, −3.96113283407428414798742466100, −2.87499964342144677143653125880, −1.54915141275242103065380355926,
0.28251602250033796718001700160, 1.68567845840308671785710877009, 3.39152055664317706220614309799, 3.99890993808702821659865255287, 4.69475199221694731285785504700, 5.96614943718274270724040725858, 6.93684498522998552795965186310, 7.61107721419777784667784811882, 8.176207757833582373105038558197, 9.096948682106048253356835412496