L(s) = 1 | − 1.73·3-s + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s + 2.99·9-s + (−0.866 − 1.5i)11-s + (−1.5 + 2.59i)13-s + (0.866 − 1.49i)15-s + 4·17-s + 6.92·19-s + (1.49 + 2.59i)21-s + (−4.33 + 7.5i)23-s + (2 + 3.46i)25-s − 5.19·27-s + (0.5 + 0.866i)29-s + (2.59 − 4.5i)31-s + ⋯ |
L(s) = 1 | − 1.00·3-s + (−0.223 + 0.387i)5-s + (−0.327 − 0.566i)7-s + 0.999·9-s + (−0.261 − 0.452i)11-s + (−0.416 + 0.720i)13-s + (0.223 − 0.387i)15-s + 0.970·17-s + 1.58·19-s + (0.327 + 0.566i)21-s + (−0.902 + 1.56i)23-s + (0.400 + 0.692i)25-s − 1.00·27-s + (0.0928 + 0.160i)29-s + (0.466 − 0.808i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.871048 + 0.317035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.871048 + 0.317035i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + (4.33 - 7.5i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 - 7.5i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.33 + 7.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + (2.59 + 4.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.33 - 7.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + (-1.5 - 2.59i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.01221121458362073905798964412, −9.857598005828976264418580724321, −9.547874672820915986684774364673, −7.68473039817180142527553158022, −7.37590504463993109679122082694, −6.17815024321373516524761835285, −5.42345236604146557386579776974, −4.23725515844151126094460315478, −3.15688185725930007471560628015, −1.14955215450611724539340734121,
0.74824771779610708114160593768, 2.64210390742601248164174795270, 4.17145312869227805869359197685, 5.23770239913816992599115931990, 5.82632683410062399018216428840, 7.01020004229341744030646703982, 7.84165846407462621901386814862, 8.920837932191506036001923355549, 10.18296665693733248169173399039, 10.29230200979393891471279956020