| L(s) = 1 | + (−0.536 − 1.65i)2-s + (4.03 − 2.92i)4-s + (−8.44 + 7.32i)5-s + 7.66·7-s + (−18.2 − 13.2i)8-s + (16.6 + 10.0i)10-s + (−6.32 − 19.4i)11-s + (−4.57 + 14.0i)13-s + (−4.11 − 12.6i)14-s + (0.205 − 0.633i)16-s + (−88.1 − 64.0i)17-s + (−79.1 − 57.4i)19-s + (−12.5 + 54.2i)20-s + (−28.7 + 20.9i)22-s + (24.8 + 76.6i)23-s + ⋯ |
| L(s) = 1 | + (−0.189 − 0.584i)2-s + (0.503 − 0.366i)4-s + (−0.755 + 0.655i)5-s + 0.413·7-s + (−0.806 − 0.585i)8-s + (0.526 + 0.316i)10-s + (−0.173 − 0.533i)11-s + (−0.0976 + 0.300i)13-s + (−0.0785 − 0.241i)14-s + (0.00321 − 0.00990i)16-s + (−1.25 − 0.914i)17-s + (−0.955 − 0.694i)19-s + (−0.140 + 0.606i)20-s + (−0.278 + 0.202i)22-s + (0.225 + 0.694i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0244647 + 0.617580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0244647 + 0.617580i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (8.44 - 7.32i)T \) |
| good | 2 | \( 1 + (0.536 + 1.65i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 7.66T + 343T^{2} \) |
| 11 | \( 1 + (6.32 + 19.4i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (4.57 - 14.0i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (88.1 + 64.0i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (79.1 + 57.4i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-24.8 - 76.6i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-47.7 + 34.7i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (142. + 103. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (71.2 - 219. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (55.4 - 170. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-386. + 280. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (417. - 303. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-176. + 543. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (16.1 + 49.7i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-446. - 324. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-298. + 216. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (125. + 386. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-529. + 384. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-341. - 247. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (131. + 404. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (477. - 347. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.30668310333699611657289573326, −10.67470734044805957644074005064, −9.503454155506654358462848865284, −8.401872318092732408809725738922, −7.13077344641402612203136193945, −6.37880819154080853577256703887, −4.77352881048150312533916923283, −3.30237476910348052260988501505, −2.13622776638131490315989483138, −0.24794733226997196657507318753,
2.02937788033314187852307672431, 3.79754340077595778229376479509, 5.00111826748413864588812616873, 6.38033718156856499974132779476, 7.38697186777595827660012247465, 8.323343105012536728380617827886, 8.854355891680226715851795448779, 10.54853616723772926714358436359, 11.32856971230492288422075961490, 12.48351558914385859724707092595
