L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 4.24i·5-s + 2.82i·8-s + 6·10-s + 6·13-s + 4.00·16-s − 4.24i·17-s − 8.48i·20-s − 12.9·25-s − 8.48i·26-s + 9.89i·29-s − 5.65i·32-s − 6·34-s − 2·37-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 1.89i·5-s + 1.00i·8-s + 1.89·10-s + 1.66·13-s + 1.00·16-s − 1.02i·17-s − 1.89i·20-s − 2.59·25-s − 1.66i·26-s + 1.83i·29-s − 1.00i·32-s − 1.02·34-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.357377706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357377706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4.24iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−9.652173553832694733360080910155, −8.801394743501445615550369381552, −7.916842880453020762746270948022, −6.98576482890637229015149049786, −6.26884477414532737926062521435, −5.31136998390671865165742300088, −4.03822964536088680837124253099, −3.24958931687781157788739063697, −2.71113413995048840570646202076, −1.40433213360464605806209832799,
0.56543345258875710026540621223, 1.64625240467082765857055783044, 3.89160305151847577774218133684, 4.16319772384089492977424346533, 5.37875314578908481358825924015, 5.77045337030886139303586056636, 6.66046356916649474593699749730, 7.937538777983804333700619065598, 8.358622308332705733988312252279, 8.873119540056533245525347234187