L(s) = 1 | + (0.326 + 1.37i)2-s + (2.18 + 2.05i)3-s + (−1.78 + 0.897i)4-s + (−1.03 + 0.772i)5-s + (−2.10 + 3.68i)6-s + (−4.52 + 2.97i)7-s + (−1.81 − 2.16i)8-s + (0.588 + 8.98i)9-s + (−1.40 − 1.17i)10-s + (−0.852 + 7.29i)11-s + (−5.75 − 1.69i)12-s + (0.717 + 2.39i)13-s + (−5.57 − 5.26i)14-s + (−3.85 − 0.436i)15-s + (2.38 − 3.20i)16-s + (12.8 + 2.26i)17-s + ⋯ |
L(s) = 1 | + (0.163 + 0.688i)2-s + (0.729 + 0.683i)3-s + (−0.446 + 0.224i)4-s + (−0.207 + 0.154i)5-s + (−0.351 + 0.613i)6-s + (−0.647 + 0.425i)7-s + (−0.227 − 0.270i)8-s + (0.0654 + 0.997i)9-s + (−0.140 − 0.117i)10-s + (−0.0774 + 0.662i)11-s + (−0.479 − 0.141i)12-s + (0.0551 + 0.184i)13-s + (−0.398 − 0.375i)14-s + (−0.257 − 0.0290i)15-s + (0.149 − 0.200i)16-s + (0.754 + 0.132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.579342 + 1.53364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.579342 + 1.53364i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.326 - 1.37i)T \) |
| 3 | \( 1 + (-2.18 - 2.05i)T \) |
good | 5 | \( 1 + (1.03 - 0.772i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (4.52 - 2.97i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (0.852 - 7.29i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-0.717 - 2.39i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-12.8 - 2.26i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (0.163 + 0.928i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-23.9 + 36.3i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-10.5 + 9.97i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (0.520 - 8.93i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-18.2 + 6.63i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-2.27 + 9.59i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-2.47 - 5.74i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-64.0 + 3.72i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-31.0 - 17.9i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (12.6 + 108. i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-55.2 - 27.7i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-8.81 + 9.34i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-9.13 + 10.8i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-31.7 + 26.6i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (76.6 - 18.1i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (0.889 + 3.75i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-7.81 - 9.31i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (25.5 - 34.2i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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
−13.14448311693762407212347997521, −12.30579385795193900956946736143, −10.78658050174289995532716025560, −9.719056128518928116633357227912, −8.929878261083549850761755479780, −7.85727171908937991134329094324, −6.75552621507181357226019924408, −5.32142272317336864980057912150, −4.10354830552362013777558628335, −2.79151174323904941281768777820,
0.965763838215728270979713613647, 2.86779025421993224346103860664, 3.85284000285783488235575952215, 5.70235756250974964381422170061, 7.08778850932232550180586294353, 8.179593776609689806941487350039, 9.232840693063506326557208986957, 10.17173568189999572413222686505, 11.45455770392296862901168594193, 12.36296033579070932868275694628