L(s) = 1 | + (0.866 − 0.5i)2-s + (1.60 + 0.923i)3-s + (0.499 − 0.866i)4-s + (−2.80 + 1.61i)5-s + 1.84·6-s − 0.999i·8-s + (0.207 + 0.358i)9-s + (−1.61 + 2.80i)10-s + (−2.04 + 2.60i)11-s + (1.60 − 0.923i)12-s + 6.10·13-s − 5.98·15-s + (−0.5 − 0.866i)16-s + (−4.06 + 7.04i)17-s + (0.358 + 0.207i)18-s + (3.30 + 5.72i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.923 + 0.533i)3-s + (0.249 − 0.433i)4-s + (−1.25 + 0.723i)5-s + 0.754·6-s − 0.353i·8-s + (0.0690 + 0.119i)9-s + (−0.511 + 0.886i)10-s + (−0.617 + 0.786i)11-s + (0.461 − 0.266i)12-s + 1.69·13-s − 1.54·15-s + (−0.125 − 0.216i)16-s + (−0.986 + 1.70i)17-s + (0.0845 + 0.0488i)18-s + (0.758 + 1.31i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.305134054\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305134054\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.04 - 2.60i)T \) |
good | 3 | \( 1 + (-1.60 - 0.923i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.80 - 1.61i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 + (4.06 - 7.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.30 - 5.72i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.65 - 4.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.32iT - 29T^{2} \) |
| 31 | \( 1 + (-2.03 - 1.17i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.64 + 4.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 3.73iT - 43T^{2} \) |
| 47 | \( 1 + (1.47 - 0.853i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.99 + 3.44i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.82 + 4.51i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.27 - 7.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.17 - 2.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + (-5.49 + 9.51i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.34 + 4.24i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 + (-4.03 + 2.33i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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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
−10.34715934733799197376026851922, −9.183376058238148155877397235485, −8.338065411408778771439385765692, −7.72881510166483725072429232484, −6.67543060669812766691711311560, −5.72747763506985006627470183824, −4.32068931045697770198065183825, −3.64157692891607798252249031345, −3.27736152786095064222703183077, −1.81143762114367535927371728006,
0.77269130955830644649792990444, 2.68555863164704608171677718251, 3.33700142314197226470292661570, 4.49931987296736069789618636712, 5.19062239057202807343025890948, 6.53735479138092404667959982935, 7.32053603189771507407690203250, 8.133435526651461663562606784315, 8.597273633498227712953109464138, 9.187763066696678451900366339668