L(s) = 1 | + (−2.66 − 0.934i)2-s + (6.25 + 4.98i)4-s + (−4.30 − 2.48i)5-s + (3.07 − 1.77i)7-s + (−12.0 − 19.1i)8-s + (9.16 + 10.6i)10-s + (22.1 + 38.4i)11-s + (30.6 − 53.1i)13-s + (−9.85 + 1.86i)14-s + (14.2 + 62.4i)16-s − 99.9i·17-s − 85.6i·19-s + (−14.5 − 37.0i)20-s + (−23.3 − 123. i)22-s + (41.3 − 71.5i)23-s + ⋯ |
L(s) = 1 | + (−0.943 − 0.330i)2-s + (0.781 + 0.623i)4-s + (−0.385 − 0.222i)5-s + (0.165 − 0.0957i)7-s + (−0.531 − 0.846i)8-s + (0.289 + 0.337i)10-s + (0.608 + 1.05i)11-s + (0.654 − 1.13i)13-s + (−0.188 + 0.0355i)14-s + (0.221 + 0.975i)16-s − 1.42i·17-s − 1.03i·19-s + (−0.162 − 0.413i)20-s + (−0.225 − 1.19i)22-s + (0.374 − 0.648i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.761949 - 0.563501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.761949 - 0.563501i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.66 + 0.934i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (4.30 + 2.48i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-3.07 + 1.77i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-22.1 - 38.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-30.6 + 53.1i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 99.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 85.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-41.3 + 71.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-152. + 88.3i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-171. - 98.9i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (38.2 + 22.0i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-67.7 + 39.0i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-229. - 398. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 290. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-147. + 256. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (247. + 428. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (19.4 + 11.2i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 304.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (964. - 556. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (400. + 694. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 346. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-291. - 504. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.62957865239235405884977723741, −11.86746822169044597539923395064, −10.81620796927825516629631673887, −9.769816059025500951294272305286, −8.717851943238992231482822395064, −7.66376021234219150435369440528, −6.58542806594224571662680867251, −4.57528212620374830262187226411, −2.79848763356262619442110851479, −0.793071946469265362648472798855,
1.48792392796386617369885102875, 3.68691826215852240460298259592, 5.82175710555337806079667516309, 6.78605178297761910634269328199, 8.190683148730734262245054978225, 8.870509924618512336141165443493, 10.21054231676006161563863743448, 11.24439690808364262376765329016, 11.94699091005385585253075002921, 13.68675547725637599043305734072