| L(s) = 1 | + (−2.97 − 2.16i)2-s + (−5.19 − 0.138i)3-s + (1.70 + 5.24i)4-s + (10.1 + 13.9i)5-s + (15.1 + 11.6i)6-s + (−16.0 + 5.21i)7-s + (−2.82 + 8.70i)8-s + (26.9 + 1.43i)9-s − 63.2i·10-s + (−36.4 − 0.331i)11-s + (−8.11 − 27.4i)12-s + (−34.3 + 47.2i)13-s + (59.0 + 19.1i)14-s + (−50.6 − 73.7i)15-s + (62.8 − 45.6i)16-s + (1.44 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.763i)2-s + (−0.999 − 0.0266i)3-s + (0.212 + 0.655i)4-s + (0.905 + 1.24i)5-s + (1.03 + 0.791i)6-s + (−0.867 + 0.281i)7-s + (−0.124 + 0.384i)8-s + (0.998 + 0.0532i)9-s − 2.00i·10-s + (−0.999 − 0.00908i)11-s + (−0.195 − 0.660i)12-s + (−0.732 + 1.00i)13-s + (1.12 + 0.366i)14-s + (−0.871 − 1.26i)15-s + (0.982 − 0.713i)16-s + (0.0206 − 0.0149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.248886 + 0.223975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248886 + 0.223975i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.19 + 0.138i)T \) |
| 11 | \( 1 + (36.4 + 0.331i)T \) |
| good | 2 | \( 1 + (2.97 + 2.16i)T + (2.47 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-10.1 - 13.9i)T + (-38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (16.0 - 5.21i)T + (277. - 201. i)T^{2} \) |
| 13 | \( 1 + (34.3 - 47.2i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 1.05i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (22.6 + 7.35i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 75.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (17.7 + 54.5i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-127. - 92.9i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (71.2 + 219. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-50.3 + 154. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 128. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-404. - 131. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (357. - 492. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-13.0 + 4.23i)T + (1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-455. - 626. i)T + (-7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 199.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-304. - 418. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-930. + 302. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (486. - 669. i)T + (-1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (443. - 322. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 399. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (717. + 521. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.05370733877396372398844004927, −15.60911287442917558147613024408, −13.98643127592948361966565959076, −12.42389852701545068155060278109, −11.10678332753906212341716755124, −10.24175415056679967595653666932, −9.477919173746900485135925859778, −7.11710470564836571197609302995, −5.74327594961245201030386400288, −2.41017671612353576450305760148,
0.45602749246438089633642093681, 5.18147525091730221737325004010, 6.46706327605445259499558642965, 8.065026540254379036135812069679, 9.636374455276602473367965145461, 10.28042566472746218113096738795, 12.60497445942427971858053687252, 13.07347674613485505495975586618, 15.51025670214511281933921698028, 16.40114295899881579965642820210
