L(s) = 1 | + (−0.134 + 1.40i)2-s + (−1.96 − 0.378i)4-s − 2.61i·5-s − 4.81i·7-s + (0.797 − 2.71i)8-s + (3.68 + 0.351i)10-s − 3.49·11-s + 1.50·13-s + (6.77 + 0.647i)14-s + (3.71 + 1.48i)16-s − 5.31·17-s + (3.83 + 2.07i)19-s + (−0.990 + 5.13i)20-s + (0.470 − 4.91i)22-s + 2.26i·23-s + ⋯ |
L(s) = 1 | + (−0.0951 + 0.995i)2-s + (−0.981 − 0.189i)4-s − 1.16i·5-s − 1.81i·7-s + (0.281 − 0.959i)8-s + (1.16 + 0.111i)10-s − 1.05·11-s + 0.416·13-s + (1.81 + 0.173i)14-s + (0.928 + 0.371i)16-s − 1.28·17-s + (0.879 + 0.475i)19-s + (−0.221 + 1.14i)20-s + (0.100 − 1.04i)22-s + 0.472i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5664794795\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5664794795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.134 - 1.40i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.83 - 2.07i)T \) |
good | 5 | \( 1 + 2.61iT - 5T^{2} \) |
| 7 | \( 1 + 4.81iT - 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 23 | \( 1 - 2.26iT - 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 3.90iT - 41T^{2} \) |
| 43 | \( 1 - 0.342T + 43T^{2} \) |
| 47 | \( 1 - 0.679iT - 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 9.15iT - 59T^{2} \) |
| 61 | \( 1 + 2.65iT - 61T^{2} \) |
| 67 | \( 1 - 7.37iT - 67T^{2} \) |
| 71 | \( 1 + 9.74T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 + 5.21T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 + 14.0iT - 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−8.983266183766584503727383762873, −8.480899368931237941988984307905, −7.40923907352441780310869128446, −7.22563826896069079313452178399, −5.91797529688900691801630974174, −5.08053000722324434071804076017, −4.37442827709483088381406627764, −3.58696864066861334947060815349, −1.38590981939254705936213056397, −0.24623034527558878808964666786,
2.08126674808360984361486238982, 2.66516699624423946853476069557, 3.42205983730426477376697792489, 4.86730363845593775443073586170, 5.57037384356145454880597766840, 6.54155220418629533608472796683, 7.63016174661186637271207093387, 8.641869968943266218018352367621, 9.053621661624419409147764949341, 10.04354580889432282845954658234