L(s) = 1 | + (0.987 − 1.42i)3-s + (−1.38 − 2.40i)5-s + (1.74 + 1.99i)7-s + (−1.04 − 2.81i)9-s + (1.71 − 2.97i)11-s + (−0.429 + 0.743i)13-s + (−4.78 − 0.398i)15-s + (−0.405 − 0.701i)17-s + (0.750 − 1.29i)19-s + (4.55 − 0.508i)21-s + (−3.82 − 6.62i)23-s + (−1.34 + 2.32i)25-s + (−5.03 − 1.28i)27-s + (3.99 + 6.92i)29-s − 7.21·31-s + ⋯ |
L(s) = 1 | + (0.570 − 0.821i)3-s + (−0.619 − 1.07i)5-s + (0.657 + 0.753i)7-s + (−0.349 − 0.936i)9-s + (0.518 − 0.898i)11-s + (−0.119 + 0.206i)13-s + (−1.23 − 0.102i)15-s + (−0.0982 − 0.170i)17-s + (0.172 − 0.298i)19-s + (0.993 − 0.110i)21-s + (−0.797 − 1.38i)23-s + (−0.268 + 0.464i)25-s + (−0.969 − 0.246i)27-s + (0.742 + 1.28i)29-s − 1.29·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.948894 - 1.23125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.948894 - 1.23125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.987 + 1.42i)T \) |
| 7 | \( 1 + (-1.74 - 1.99i)T \) |
good | 5 | \( 1 + (1.38 + 2.40i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.71 + 2.97i)T + (-5.5 - 9.52i)T^{2} \) |
| 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 | \( 1 + (3.82 + 6.62i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.99 - 6.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 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 | \( 1 + 0.615T + 47T^{2} \) |
| 53 | \( 1 + (-6.31 - 10.9i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + (4.16 + 7.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 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
−10.92245765785808942604080407230, −9.300421697643306828213432178792, −8.625749961312199433329866848495, −8.283085409593697758591899649766, −7.16192690187227225429097880445, −6.04562509393407548051040642206, −4.95467137753257749584756830226, −3.77936083770266014749545067118, −2.34675239665914043047620257995, −0.922910818371482538023012523772,
2.12424964809972483088697458627, 3.60841556372745647934052853815, 4.12162025219419529537494253649, 5.36744567909858966849497989427, 6.90184174945286383708427253744, 7.62225026607865136308453046824, 8.342831826851824307440794997823, 9.758225093934091886178448293738, 10.10953657109115255948418018793, 11.23934381100756757730803752279