L(s) = 1 | + (−0.360 − 1.96i)2-s + (2.49 − 1.03i)3-s + (−3.73 + 1.41i)4-s + (−0.452 − 0.187i)5-s + (−2.93 − 4.53i)6-s + (0.429 − 0.429i)7-s + (4.14 + 6.84i)8-s + (−1.19 + 1.19i)9-s + (−0.205 + 0.957i)10-s + (17.3 + 7.18i)11-s + (−7.86 + 7.41i)12-s + (−19.9 + 8.26i)13-s + (−1.00 − 0.690i)14-s − 1.32·15-s + (11.9 − 10.6i)16-s − 13.5i·17-s + ⋯ |
L(s) = 1 | + (−0.180 − 0.983i)2-s + (0.832 − 0.344i)3-s + (−0.934 + 0.354i)4-s + (−0.0904 − 0.0374i)5-s + (−0.489 − 0.756i)6-s + (0.0614 − 0.0614i)7-s + (0.517 + 0.855i)8-s + (−0.133 + 0.133i)9-s + (−0.0205 + 0.0957i)10-s + (1.57 + 0.652i)11-s + (−0.655 + 0.617i)12-s + (−1.53 + 0.635i)13-s + (−0.0714 − 0.0493i)14-s − 0.0882·15-s + (0.747 − 0.663i)16-s − 0.799i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.844035 - 0.610648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.844035 - 0.610648i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.360 + 1.96i)T \) |
good | 3 | \( 1 + (-2.49 + 1.03i)T + (6.36 - 6.36i)T^{2} \) |
| 5 | \( 1 + (0.452 + 0.187i)T + (17.6 + 17.6i)T^{2} \) |
| 7 | \( 1 + (-0.429 + 0.429i)T - 49iT^{2} \) |
| 11 | \( 1 + (-17.3 - 7.18i)T + (85.5 + 85.5i)T^{2} \) |
| 13 | \( 1 + (19.9 - 8.26i)T + (119. - 119. i)T^{2} \) |
| 17 | \( 1 + 13.5iT - 289T^{2} \) |
| 19 | \( 1 + (3.45 + 8.34i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (16.8 + 16.8i)T + 529iT^{2} \) |
| 29 | \( 1 + (-13.8 - 33.4i)T + (-594. + 594. i)T^{2} \) |
| 31 | \( 1 + 24.5iT - 961T^{2} \) |
| 37 | \( 1 + (-9.89 - 4.09i)T + (968. + 968. i)T^{2} \) |
| 41 | \( 1 + (-14.4 + 14.4i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-17.8 - 7.39i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 - 43.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-28.0 + 67.7i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-1.70 + 4.10i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-3.53 - 8.53i)T + (-2.63e3 + 2.63e3i)T^{2} \) |
| 67 | \( 1 + (0.300 - 0.124i)T + (3.17e3 - 3.17e3i)T^{2} \) |
| 71 | \( 1 + (29.0 - 29.0i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (68.2 - 68.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 67.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (16.4 + 39.5i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (45.3 + 45.3i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + 119.T + 9.40e3T^{2} \) |
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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
−16.71963171206596534686266141715, −14.52545261970177613749119107524, −14.05074454321922415567262874089, −12.48719797132526451463916850218, −11.59174082864454174303821665671, −9.804317309264955915405848851760, −8.840451674379023619253627078378, −7.32357019631110641216283871430, −4.39978942781689190287612079515, −2.32548883631757481471812895439,
3.90255425921258343796541611581, 6.00089047485615853975053698047, 7.73477348787961148403041400662, 8.934926843954843830041571193246, 9.943819741442571779654539249601, 12.06823175869098702527451604462, 13.81076019672503983454982379521, 14.65047172215173706643027273445, 15.37898635483016397433316228948, 16.89150639141384729422954338725