L(s) = 1 | + (0.0340 − 0.0589i)2-s + (0.997 + 1.72i)4-s + (−1.52 − 2.64i)5-s + (−2.60 + 0.453i)7-s + 0.271·8-s − 0.208·10-s + 4.35·11-s + (−1.79 − 3.12i)13-s + (−0.0619 + 0.169i)14-s + (−1.98 + 3.44i)16-s + (−1.76 − 3.05i)17-s + 6.90·19-s + (3.05 − 5.28i)20-s + (0.148 − 0.256i)22-s + (1.66 − 2.88i)23-s + ⋯ |
L(s) = 1 | + (0.0240 − 0.0416i)2-s + (0.498 + 0.864i)4-s + (−0.684 − 1.18i)5-s + (−0.985 + 0.171i)7-s + 0.0961·8-s − 0.0658·10-s + 1.31·11-s + (−0.499 − 0.866i)13-s + (−0.0165 + 0.0451i)14-s + (−0.496 + 0.860i)16-s + (−0.427 − 0.741i)17-s + 1.58·19-s + (0.682 − 1.18i)20-s + (0.0316 − 0.0547i)22-s + (0.347 − 0.602i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06589 - 0.725012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06589 - 0.725012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.60 - 0.453i)T \) |
| 13 | \( 1 + (1.79 + 3.12i)T \) |
good | 2 | \( 1 + (-0.0340 + 0.0589i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.52 + 2.64i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 17 | \( 1 + (1.76 + 3.05i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 + (-1.66 + 2.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.95 + 8.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.62 + 8.00i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0545 + 0.0944i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.76 + 3.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.844 - 1.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.28 + 2.21i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.65 - 4.60i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.77 - 6.54i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 0.680T + 67T^{2} \) |
| 71 | \( 1 + (-2.61 + 4.53i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.75 + 3.04i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.85 - 8.40i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 + (3.85 - 6.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.86 - 6.69i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.752776452589323738564182223341, −9.267872171455672719212761744211, −8.320283177563942108947277153034, −7.57620017151408344512772735186, −6.75837267232871166629272079239, −5.67734733718088908195354954070, −4.43492018011108568311360928521, −3.65095561444314718746781074692, −2.59079724979655038206422614439, −0.66457634493936205488864598850,
1.48699418714608180642156120355, 3.04345406758176921212306571544, 3.75698471645858965059414192780, 5.15413380478920425919214720322, 6.43023517758716407799076720225, 6.82892669650200699816987341010, 7.37029879776543769440502329661, 8.961965806968757438794927285904, 9.680305836769620435396841709642, 10.37467307426398810005389794992