L(s) = 1 | + 2-s + (0.618 + 1.61i)3-s + 4-s + (0.618 + 1.61i)6-s + (0.381 + 2.61i)7-s + 8-s + (−2.23 + 2.00i)9-s − 4.47i·11-s + (0.618 + 1.61i)12-s + 1.23·13-s + (0.381 + 2.61i)14-s + 16-s + 5.23i·17-s + (−2.23 + 2.00i)18-s + 8.47i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.356 + 0.934i)3-s + 0.5·4-s + (0.252 + 0.660i)6-s + (0.144 + 0.989i)7-s + 0.353·8-s + (−0.745 + 0.666i)9-s − 1.34i·11-s + (0.178 + 0.467i)12-s + 0.342·13-s + (0.102 + 0.699i)14-s + 0.250·16-s + 1.26i·17-s + (−0.527 + 0.471i)18-s + 1.94i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0460 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0460 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.738780289\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.738780289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.381 - 2.61i)T \) |
good | 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.23iT - 17T^{2} \) |
| 19 | \( 1 - 8.47iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.70iT - 29T^{2} \) |
| 31 | \( 1 - 2.76iT - 31T^{2} \) |
| 37 | \( 1 - 0.763iT - 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 4.94iT - 43T^{2} \) |
| 47 | \( 1 + 6.47iT - 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 7.23iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 7.23iT - 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 - 0.763T + 97T^{2} \) |
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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
−10.25636842846039878044010349089, −9.289202205164036263625394891561, −8.302702282120593364786538459486, −8.079211251067449796923342831259, −6.19351256455667677641751673872, −5.86276503884085867126285581071, −4.92613237736275825366379992445, −3.75557977220978641492751469482, −3.21633726460588618721200352800, −1.92054078373990136184851912075,
0.980090286496742946707337151453, 2.31363722480896651569688592899, 3.26890809992590836219366957442, 4.51672718268051537116152611533, 5.20569160946380895495173071772, 6.70138906176329705354920358828, 7.06110224803891365875955789111, 7.61369398972896546991146891092, 8.876377585119946391312235864699, 9.595439635067450937838775202787