| L(s) = 1 | + (4.39 − 2.53i)2-s + (0.143 + 5.19i)3-s + (8.88 − 15.3i)4-s + 8.42·5-s + (13.8 + 22.4i)6-s + (−13.4 + 12.7i)7-s − 49.6i·8-s + (−26.9 + 1.49i)9-s + (37.0 − 21.3i)10-s − 45.8i·11-s + (81.2 + 43.9i)12-s + (−18.3 + 10.6i)13-s + (−26.8 + 90.1i)14-s + (1.20 + 43.7i)15-s + (−54.8 − 95.0i)16-s + (8.26 + 14.3i)17-s + ⋯ |
| L(s) = 1 | + (1.55 − 0.897i)2-s + (0.0276 + 0.999i)3-s + (1.11 − 1.92i)4-s + 0.753·5-s + (0.940 + 1.52i)6-s + (−0.726 + 0.687i)7-s − 2.19i·8-s + (−0.998 + 0.0552i)9-s + (1.17 − 0.676i)10-s − 1.25i·11-s + (1.95 + 1.05i)12-s + (−0.392 + 0.226i)13-s + (−0.512 + 1.72i)14-s + (0.0208 + 0.753i)15-s + (−0.857 − 1.48i)16-s + (0.117 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.01773 - 0.827626i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.01773 - 0.827626i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.143 - 5.19i)T \) |
| 7 | \( 1 + (13.4 - 12.7i)T \) |
| good | 2 | \( 1 + (-4.39 + 2.53i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 8.42T + 125T^{2} \) |
| 11 | \( 1 + 45.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (18.3 - 10.6i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-8.26 - 14.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (49.7 + 28.7i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 193. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (47.1 + 27.2i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-254. - 147. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-185. + 320. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (166. + 287. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-209. - 363. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-275. + 159. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (149. - 258. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (195. - 113. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-49.8 + 86.3i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 176. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-142. + 82.2i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (187. + 324. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (457. - 792. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (67.8 - 117. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-377. - 217. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.05397703335558979791427620728, −13.45001361903089324923088536240, −12.14773324835995045696105143485, −11.15801103853807444215446694127, −10.11481687379589314826319741135, −9.073419797475539033243762417589, −6.06480863454120652878235475076, −5.42319372006970072289738760208, −3.78743845002564362012142248924, −2.58293603377393909626906312490,
2.60434401935367961473336630721, 4.54667993663911648068857909833, 6.13803514978220340075002944637, 6.81264590184761366349597216712, 7.944297131160512093364907601194, 10.01398731915145276985670571424, 11.96002125056504811316327973832, 12.83458517114090764425990253019, 13.41197236953942917844085064809, 14.35917254776531703831727778621
