| L(s) = 1 | + (−1.17 − 1.62i)2-s + (−0.157 + 2.99i)3-s + (−0.00519 + 0.0159i)4-s + (3.22 − 4.43i)5-s + (5.04 − 3.27i)6-s + (−1.72 + 5.31i)7-s + (−7.59 + 2.46i)8-s + (−8.95 − 0.942i)9-s − 10.9·10-s + (−0.0470 − 0.0180i)12-s + (−7.86 + 5.71i)13-s + (10.6 − 3.46i)14-s + (12.7 + 10.3i)15-s + (12.9 + 9.44i)16-s + (−10.5 + 14.4i)17-s + (9.01 + 15.6i)18-s + ⋯ |
| L(s) = 1 | + (−0.589 − 0.810i)2-s + (−0.0524 + 0.998i)3-s + (−0.00129 + 0.00399i)4-s + (0.644 − 0.887i)5-s + (0.840 − 0.545i)6-s + (−0.246 + 0.759i)7-s + (−0.949 + 0.308i)8-s + (−0.994 − 0.104i)9-s − 1.09·10-s + (−0.00392 − 0.00150i)12-s + (−0.604 + 0.439i)13-s + (0.761 − 0.247i)14-s + (0.852 + 0.690i)15-s + (0.812 + 0.590i)16-s + (−0.617 + 0.850i)17-s + (0.500 + 0.867i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.156492 + 0.259162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.156492 + 0.259162i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.157 - 2.99i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.17 + 1.62i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (-3.22 + 4.43i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (1.72 - 5.31i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (7.86 - 5.71i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (10.5 - 14.4i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (5.76 + 17.7i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + 12.3iT - 529T^{2} \) |
| 29 | \( 1 + (2.35 + 0.765i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (39.8 - 28.9i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (12.1 - 37.4i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (53.7 - 17.4i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 43.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (54.9 - 17.8i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-25.3 - 34.8i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (85.6 + 27.8i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (24.9 + 18.1i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 34.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (22.0 - 30.4i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-3.74 + 11.5i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (51.0 - 37.1i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-5.70 + 7.84i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 34.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-30.5 + 22.2i)T + (2.90e3 - 8.94e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.26069351991037577132631871434, −10.49830914937018623509787002640, −9.614683872346082558454126404927, −9.018666842030557146485983918890, −8.560400024374605939754466634428, −6.45722894542464225743291154474, −5.48731007885794494351291525059, −4.62493883782343681414020698833, −2.98255112159218567279175695399, −1.80608820250242311271402594827,
0.15276416005495285600017210460, 2.22753699456408722657177997450, 3.45018346947108581264645354885, 5.57879990054309019610212510467, 6.45536265780989864590611319947, 7.22268154482348148838939744355, 7.66796786649206731617596322059, 8.875917133999220583615748318144, 9.855460859752720350642488877673, 10.81651604997623993272125370623
