| L(s) = 1 | + (3.16 − 2.29i)2-s + (2.24 − 6.90i)4-s + (11.0 − 1.45i)5-s + 22.0·7-s + (0.889 + 2.73i)8-s + (31.6 − 30.0i)10-s + (−31.5 + 22.9i)11-s + (55.3 + 40.2i)13-s + (69.8 − 50.7i)14-s + (56.1 + 40.7i)16-s + (−31.1 − 95.9i)17-s + (−29.0 − 89.4i)19-s + (14.8 − 79.8i)20-s + (−47.1 + 145. i)22-s + (−130. + 94.8i)23-s + ⋯ |
| L(s) = 1 | + (1.11 − 0.811i)2-s + (0.280 − 0.863i)4-s + (0.991 − 0.130i)5-s + 1.19·7-s + (0.0393 + 0.120i)8-s + (1.00 − 0.950i)10-s + (−0.866 + 0.629i)11-s + (1.18 + 0.857i)13-s + (1.33 − 0.968i)14-s + (0.876 + 0.636i)16-s + (−0.444 − 1.36i)17-s + (−0.350 − 1.07i)19-s + (0.165 − 0.892i)20-s + (−0.456 + 1.40i)22-s + (−1.18 + 0.859i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.70232 - 1.72017i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.70232 - 1.72017i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-11.0 + 1.45i)T \) |
| good | 2 | \( 1 + (-3.16 + 2.29i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 - 22.0T + 343T^{2} \) |
| 11 | \( 1 + (31.5 - 22.9i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-55.3 - 40.2i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (31.1 + 95.9i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (29.0 + 89.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (130. - 94.8i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-15.8 + 48.7i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (80.4 + 247. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-70.1 - 50.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (42.4 + 30.8i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 53.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (74.7 - 230. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (18.2 - 56.2i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (524. + 380. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (530. - 385. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-240. - 740. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (69.2 - 213. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (281. - 204. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-49.9 + 153. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 37.9i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (375. - 272. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-177. + 546. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.46169714916263230801024228166, −11.22878402854762406692691822175, −9.978810994353100170189108239742, −8.866852866807031241657911144304, −7.62405527724986132855941158037, −6.07753130083841152850647786299, −5.02898155264191740257611797336, −4.32149280902110219600703607473, −2.54656186751649603712789389532, −1.67776743644336924469897645418,
1.65004403670505469822532792693, 3.48221007212069013485701021427, 4.82018699560941492423315499022, 5.80408649610626759641658588252, 6.29590570993613604237782298543, 7.917467283447715336906496505054, 8.541562240239250767046729281526, 10.43210866144950061546339400185, 10.71330098835840673346638108185, 12.40229800713572631658848857908
