| L(s) = 1 | + (0.361 + 0.142i)2-s + (2.98 + 4.24i)3-s + (−5.75 − 5.33i)4-s + (16.3 + 11.1i)5-s + (0.478 + 1.96i)6-s + (18.4 − 1.10i)7-s + (−2.67 − 5.55i)8-s + (−9.12 + 25.4i)9-s + (4.32 + 6.34i)10-s + (−6.42 − 42.6i)11-s + (5.48 − 40.4i)12-s + (−40.4 + 32.2i)13-s + (6.84 + 2.22i)14-s + (1.50 + 102. i)15-s + (4.51 + 60.2i)16-s + (83.1 + 25.6i)17-s + ⋯ |
| L(s) = 1 | + (0.127 + 0.0502i)2-s + (0.575 + 0.817i)3-s + (−0.719 − 0.667i)4-s + (1.46 + 0.995i)5-s + (0.0325 + 0.133i)6-s + (0.998 − 0.0597i)7-s + (−0.118 − 0.245i)8-s + (−0.337 + 0.941i)9-s + (0.136 + 0.200i)10-s + (−0.176 − 1.16i)11-s + (0.131 − 0.972i)12-s + (−0.863 + 0.688i)13-s + (0.130 + 0.0424i)14-s + (0.0259 + 1.76i)15-s + (0.0705 + 0.941i)16-s + (1.18 + 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.11454 + 1.16968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11454 + 1.16968i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.98 - 4.24i)T \) |
| 7 | \( 1 + (-18.4 + 1.10i)T \) |
| good | 2 | \( 1 + (-0.361 - 0.142i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-16.3 - 11.1i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (6.42 + 42.6i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (40.4 - 32.2i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-83.1 - 25.6i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-56.0 + 32.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-36.9 - 119. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (116. - 26.5i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (169. + 98.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-167. + 155. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-147. + 70.8i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (420. + 202. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-139. + 354. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-21.0 + 22.6i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (85.5 - 58.2i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (341. + 367. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-246. + 427. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (218. + 49.7i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (115. - 45.1i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (343. + 594. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-314. + 394. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-216. - 32.6i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 321. iT - 9.12e5T^{2} \) |
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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
−13.41720402635790935142932969577, −11.30440964332036149149684077349, −10.51637730086289217086269377642, −9.663473128854561747506947355139, −9.012777927651310114691033700197, −7.52467960781439500024270414412, −5.73418211265660694799339762362, −5.19443216583764433141048458053, −3.48466747810854774487839885625, −1.89018083055762576480878523514,
1.29202702167883801744171518834, 2.64121460774211735336833671495, 4.72314690838651531239618689091, 5.54551146870413177282205662570, 7.42916782291630792438653142204, 8.192707907956745798169664863694, 9.273935824420433481273197350410, 9.935733412197358580607721617263, 12.02971405388064235888112838255, 12.63834837019130103086954291911
