| L(s) = 1 | + (4.87 + 1.91i)2-s + (−5.14 − 0.739i)3-s + (14.2 + 13.1i)4-s + (−3.32 − 2.26i)5-s + (−23.6 − 13.4i)6-s + (1.07 + 18.4i)7-s + (25.8 + 53.6i)8-s + (25.9 + 7.60i)9-s + (−11.8 − 17.3i)10-s + (6.81 + 45.2i)11-s + (−63.3 − 78.3i)12-s + (−16.7 + 13.3i)13-s + (−30.0 + 92.1i)14-s + (15.4 + 14.1i)15-s + (11.7 + 156. i)16-s + (−13.5 − 4.18i)17-s + ⋯ |
| L(s) = 1 | + (1.72 + 0.675i)2-s + (−0.989 − 0.142i)3-s + (1.77 + 1.64i)4-s + (−0.297 − 0.202i)5-s + (−1.60 − 0.914i)6-s + (0.0582 + 0.998i)7-s + (1.14 + 2.37i)8-s + (0.959 + 0.281i)9-s + (−0.374 − 0.549i)10-s + (0.186 + 1.23i)11-s + (−1.52 − 1.88i)12-s + (−0.356 + 0.284i)13-s + (−0.574 + 1.75i)14-s + (0.265 + 0.242i)15-s + (0.183 + 2.44i)16-s + (−0.193 − 0.0596i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.90638 + 2.45860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90638 + 2.45860i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.14 + 0.739i)T \) |
| 7 | \( 1 + (-1.07 - 18.4i)T \) |
| good | 2 | \( 1 + (-4.87 - 1.91i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (3.32 + 2.26i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-6.81 - 45.2i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (16.7 - 13.3i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (13.5 + 4.18i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-48.0 + 27.7i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (48.9 + 158. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-220. + 50.2i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-262. - 151. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (106. - 98.3i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-177. + 85.3i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (93.3 + 44.9i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (90.1 - 229. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-345. + 372. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-343. + 234. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (393. + 424. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (293. - 509. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-222. - 50.8i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (474. - 186. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-349. - 605. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-790. + 991. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (806. + 121. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 347. iT - 9.12e5T^{2} \) |
| 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
−12.60736251772406904162092639563, −12.16748352906755267573779695476, −11.65627951541052023093983091615, −10.10612273523863090686805945359, −8.226879718219928510238448355081, −6.91880104317839052063814856343, −6.26023936139550774307636819373, −4.94490915415665505657445427287, −4.46154511158714898598338279253, −2.43541061591533509644131618743,
1.07423888113075465779865851628, 3.31571369590130435454369825546, 4.25873410512482836409017862465, 5.43231911244114084358721651657, 6.34289984975587282903859620288, 7.50701469090066170856048657094, 9.950920465893776442002282889911, 10.76410412957185632520283497088, 11.53659090351226861985391335050, 12.11432922724641402220044989801
