L(s) = 1 | + (0.612 + 1.27i)2-s + (−1.25 + 1.56i)4-s + 4.23i·5-s − 3.84i·7-s + (−2.75 − 0.637i)8-s + (−5.39 + 2.59i)10-s − 2.95·11-s + 1.09·13-s + (4.90 − 2.35i)14-s + (−0.873 − 3.90i)16-s − 2.81·17-s + (−0.124 + 4.35i)19-s + (−6.61 − 5.29i)20-s + (−1.80 − 3.76i)22-s − 2.47i·23-s + ⋯ |
L(s) = 1 | + (0.432 + 0.901i)2-s + (−0.625 + 0.780i)4-s + 1.89i·5-s − 1.45i·7-s + (−0.974 − 0.225i)8-s + (−1.70 + 0.819i)10-s − 0.889·11-s + 0.304·13-s + (1.31 − 0.629i)14-s + (−0.218 − 0.975i)16-s − 0.682·17-s + (−0.0285 + 0.999i)19-s + (−1.47 − 1.18i)20-s + (−0.385 − 0.802i)22-s − 0.516i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4892987124\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4892987124\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.612 - 1.27i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.124 - 4.35i)T \) |
good | 5 | \( 1 - 4.23iT - 5T^{2} \) |
| 7 | \( 1 + 3.84iT - 7T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 23 | \( 1 + 2.47iT - 23T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 0.594T + 37T^{2} \) |
| 41 | \( 1 - 5.98iT - 41T^{2} \) |
| 43 | \( 1 - 8.62T + 43T^{2} \) |
| 47 | \( 1 + 8.38iT - 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 + 7.19iT - 59T^{2} \) |
| 61 | \( 1 + 1.92iT - 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 - 2.10T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.29T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 - 14.7iT - 89T^{2} \) |
| 97 | \( 1 + 4.36iT - 97T^{2} \) |
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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
−10.29531953182994568489288735238, −9.410314216797980742047007310812, −8.034209488222514709207823646997, −7.54118087864854873340237623152, −6.92302531142940117853669623364, −6.29014115382645295940114921199, −5.34310027904456711106142861283, −3.93744779601402914884810053622, −3.60830283741117259721388194173, −2.38162041923363424492520152840,
0.16245662681405339205502624924, 1.68001071169908713618523187587, 2.51488225632422082538652228852, 3.88540574955104736998355987959, 4.86501378215815455378494616331, 5.41929111077268269679961139597, 5.94231713911343195141014991490, 7.67090794790649739322970182993, 8.689670764073628312542476570196, 9.123827823888281187510093555324