L(s) = 1 | + (−0.385 + 1.68i)3-s + (1.15 − 2.00i)5-s + (−0.5 − 0.866i)7-s + (−2.70 − 1.30i)9-s + (0.655 + 1.13i)11-s + (1.93 − 3.34i)13-s + (2.93 + 2.72i)15-s − 0.326·17-s − 3.08·19-s + (1.65 − 0.510i)21-s + (1.81 − 3.15i)23-s + (−0.169 − 0.293i)25-s + (3.24 − 4.05i)27-s + (−4.75 − 8.22i)29-s + (3.24 − 5.62i)31-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)3-s + (0.516 − 0.894i)5-s + (−0.188 − 0.327i)7-s + (−0.900 − 0.434i)9-s + (0.197 + 0.342i)11-s + (0.536 − 0.928i)13-s + (0.757 + 0.703i)15-s − 0.0792·17-s − 0.707·19-s + (0.361 − 0.111i)21-s + (0.379 − 0.656i)23-s + (−0.0338 − 0.0586i)25-s + (0.624 − 0.781i)27-s + (−0.882 − 1.52i)29-s + (0.583 − 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.379795774\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.379795774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.385 - 1.68i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.15 + 2.00i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.655 - 1.13i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.93 + 3.34i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.326T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 + (-1.81 + 3.15i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.75 + 8.22i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + (-4.74 + 8.22i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0493 - 0.0855i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.108 - 0.187i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + (1.22 - 2.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.68 - 13.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 5.99T + 73T^{2} \) |
| 79 | \( 1 + (7.29 + 12.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.04 + 5.26i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + (-2.99 - 5.18i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.984412851887952790588327848854, −9.008476050614108434689920798104, −8.569445872025329596448125435209, −7.39959085492892927618080265475, −6.07542337613324953622643041757, −5.57513502041377734039527921512, −4.50209264408797911020956276671, −3.85083996976866560884685640932, −2.44180099203689080766932613397, −0.67698163222371326710688079041,
1.47307077394530054384548780923, 2.52434041632518813850684728423, 3.54494205123792899073369566456, 5.09791450443251215935423290310, 6.10862301605508995306014677555, 6.63511619856218424451507094305, 7.28132403833172272479586015128, 8.487634756155246330866684579669, 9.062055853826327389299100822160, 10.18898225517629911103656483987