| L(s) = 1 | + (−3.52 − 1.38i)2-s + (−5.17 − 0.479i)3-s + (4.63 + 4.30i)4-s + (1.95 + 1.33i)5-s + (17.5 + 8.84i)6-s + (−18.0 − 4.32i)7-s + (2.74 + 5.70i)8-s + (26.5 + 4.96i)9-s + (−5.03 − 7.38i)10-s + (8.14 + 54.0i)11-s + (−21.9 − 24.4i)12-s + (20.4 − 16.2i)13-s + (57.4 + 40.1i)14-s + (−9.45 − 7.82i)15-s + (−5.57 − 74.3i)16-s + (60.1 + 18.5i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 − 0.488i)2-s + (−0.995 − 0.0922i)3-s + (0.579 + 0.538i)4-s + (0.174 + 0.119i)5-s + (1.19 + 0.601i)6-s + (−0.972 − 0.233i)7-s + (0.121 + 0.251i)8-s + (0.982 + 0.183i)9-s + (−0.159 − 0.233i)10-s + (0.223 + 1.48i)11-s + (−0.527 − 0.589i)12-s + (0.435 − 0.347i)13-s + (1.09 + 0.766i)14-s + (−0.162 − 0.134i)15-s + (−0.0870 − 1.16i)16-s + (0.858 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.218621 - 0.292068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218621 - 0.292068i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.17 + 0.479i)T \) |
| 7 | \( 1 + (18.0 + 4.32i)T \) |
| good | 2 | \( 1 + (3.52 + 1.38i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-1.95 - 1.33i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-8.14 - 54.0i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-20.4 + 16.2i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-60.1 - 18.5i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (90.7 - 52.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (51.2 + 166. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (9.40 - 2.14i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (43.4 + 25.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-166. + 154. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (71.2 - 34.3i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-213. - 102. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-190. + 485. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (12.7 - 13.7i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-27.5 + 18.7i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (511. + 551. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-60.1 + 104. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-1.09e3 - 250. i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-479. + 188. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (235. + 407. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (853. - 1.07e3i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (570. + 85.9i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.58e3iT - 9.12e5T^{2} \) |
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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
−12.30557572246591646823536780687, −10.87210826023827818505449690183, −10.16629950018768629644075863064, −9.670970755356887694260423825081, −8.185875063128530415942351813478, −6.99996698613910547566315321555, −5.95277468298369894153736673328, −4.27728308725119151134054093828, −2.03763875911903752212047627720, −0.39707294254472577628917581201,
1.00435122548440857891528476208, 3.72901336922398172925475967201, 5.73469557285313369533197763878, 6.42544666296082915525719702407, 7.60385202928139637547336462292, 8.961794265023520515296299168830, 9.609417331018632782556010075039, 10.68903822131043289501868219409, 11.55926869152244196048406985651, 12.84399741517286070839441648059
