| L(s) = 1 | + (3.31 + 1.29i)2-s + (−2.61 + 4.49i)3-s + (3.41 + 3.16i)4-s + (12.8 + 8.78i)5-s + (−14.4 + 11.4i)6-s + (−5.96 + 17.5i)7-s + (−5.16 − 10.7i)8-s + (−13.3 − 23.4i)9-s + (31.2 + 45.8i)10-s + (3.82 + 25.3i)11-s + (−23.1 + 7.04i)12-s + (−7.06 + 5.63i)13-s + (−42.5 + 50.3i)14-s + (−73.1 + 34.8i)15-s + (−5.94 − 79.3i)16-s + (31.1 + 9.61i)17-s + ⋯ |
| L(s) = 1 | + (1.17 + 0.459i)2-s + (−0.503 + 0.864i)3-s + (0.426 + 0.395i)4-s + (1.15 + 0.785i)5-s + (−0.986 + 0.780i)6-s + (−0.321 + 0.946i)7-s + (−0.228 − 0.473i)8-s + (−0.493 − 0.869i)9-s + (0.988 + 1.44i)10-s + (0.104 + 0.694i)11-s + (−0.556 + 0.169i)12-s + (−0.150 + 0.120i)13-s + (−0.811 + 0.960i)14-s + (−1.25 + 0.600i)15-s + (−0.0929 − 1.23i)16-s + (0.444 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.16797 + 2.40269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16797 + 2.40269i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.61 - 4.49i)T \) |
| 7 | \( 1 + (5.96 - 17.5i)T \) |
| good | 2 | \( 1 + (-3.31 - 1.29i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-12.8 - 8.78i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-3.82 - 25.3i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (7.06 - 5.63i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-31.1 - 9.61i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (15.4 - 8.92i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-24.5 - 79.4i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (160. - 36.5i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-52.5 - 30.3i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-223. + 206. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-440. + 212. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-470. - 226. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-136. + 347. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (70.1 - 75.5i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-394. + 268. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-86.5 - 93.2i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (216. - 375. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (269. + 61.6i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-206. + 81.0i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-51.2 - 88.7i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (608. - 763. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (878. + 132. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.13e3iT - 9.12e5T^{2} \) |
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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
−12.98681391957803892532008006417, −12.20973398337060233535934211808, −10.97432532285955892232612803874, −9.745885746483147011364471650788, −9.327498751179829370558408465903, −7.02465344973469867011925861689, −5.87700936064133598855996928213, −5.54239406140994037663604611015, −4.07622850451995634518237359505, −2.65802870959656763947704501630,
1.01429822342240953827323199052, 2.61384955298989094314990950750, 4.38423326129564954147848884863, 5.57836731485995696190555427000, 6.27902384000223405439920104126, 7.82682604727276807283737434696, 9.196420456633897586084735147655, 10.57668365280860358227505800087, 11.50511116825657722416548653089, 12.72653235030840271865188207789
