L(s) = 1 | + (−1.68 − 2.92i)2-s + (−1.68 + 2.92i)4-s + 9.74·5-s + (0.158 + 18.5i)7-s − 15.5·8-s + (−16.4 − 28.4i)10-s − 38.9·11-s + (−31.2 − 54.1i)13-s + (53.8 − 31.6i)14-s + (39.8 + 68.9i)16-s + (−63.3 − 109. i)17-s + (22.7 − 39.3i)19-s + (−16.4 + 28.5i)20-s + (65.6 + 113. i)22-s − 153.·23-s + ⋯ |
L(s) = 1 | + (−0.596 − 1.03i)2-s + (−0.211 + 0.365i)4-s + 0.871·5-s + (0.00854 + 0.999i)7-s − 0.688·8-s + (−0.519 − 0.900i)10-s − 1.06·11-s + (−0.667 − 1.15i)13-s + (1.02 − 0.605i)14-s + (0.621 + 1.07i)16-s + (−0.903 − 1.56i)17-s + (0.274 − 0.475i)19-s + (−0.184 + 0.318i)20-s + (0.636 + 1.10i)22-s − 1.39·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.136427 + 0.428702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.136427 + 0.428702i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.158 - 18.5i)T \) |
good | 2 | \( 1 + (1.68 + 2.92i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 9.74T + 125T^{2} \) |
| 11 | \( 1 + 38.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + (31.2 + 54.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (63.3 + 109. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-22.7 + 39.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (27.4 - 47.5i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (54.8 - 95.0i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-144. + 250. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-11.3 - 19.5i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-23.9 + 41.4i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-193. - 334. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-133. - 231. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-193. + 334. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (17.8 + 30.9i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (17.8 - 30.9i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 146.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (364. + 631. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (250. + 434. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (169. - 293. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-104. + 180. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (56.7 - 98.2i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.39873224896114066241850827041, −10.43750461615427924942225394819, −9.644660317892015383146194999587, −8.956686084344554169399148972250, −7.65383171937278806525843469923, −5.97527145474657189453786897049, −5.16768084538084428825903600889, −2.80661983807514210074634800659, −2.21076535128012499339832729415, −0.21621418732656702038164563775,
2.09605508814524559816946870011, 4.12013276004571167327305572178, 5.71477782027553849614654720576, 6.58123023260123238789645619946, 7.59840625149401740577953309012, 8.445694778662083091062753248101, 9.717749917147972239698165869952, 10.28729910676759628134244552784, 11.65753195687125272348756919416, 12.94329268727761161613818705804