| L(s) = 1 | + (−0.448 + 1.37i)2-s + (−5.12 + 0.866i)3-s + (4.76 + 3.46i)4-s + (−14.6 + 4.75i)5-s + (1.10 − 7.45i)6-s + (−7.94 + 10.9i)7-s + (−16.3 + 11.8i)8-s + (25.4 − 8.88i)9-s − 22.3i·10-s + (36.2 − 4.26i)11-s + (−27.4 − 13.6i)12-s + (21.5 + 6.99i)13-s + (−11.5 − 15.8i)14-s + (70.8 − 37.0i)15-s + (5.53 + 17.0i)16-s + (27.3 + 84.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.158 + 0.487i)2-s + (−0.985 + 0.166i)3-s + (0.596 + 0.433i)4-s + (−1.30 + 0.425i)5-s + (0.0749 − 0.507i)6-s + (−0.428 + 0.590i)7-s + (−0.720 + 0.523i)8-s + (0.944 − 0.328i)9-s − 0.705i·10-s + (0.993 − 0.116i)11-s + (−0.660 − 0.327i)12-s + (0.459 + 0.149i)13-s + (−0.219 − 0.302i)14-s + (1.21 − 0.637i)15-s + (0.0864 + 0.266i)16-s + (0.389 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.226482 + 0.640506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.226482 + 0.640506i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.12 - 0.866i)T \) |
| 11 | \( 1 + (-36.2 + 4.26i)T \) |
| good | 2 | \( 1 + (0.448 - 1.37i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (14.6 - 4.75i)T + (101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (7.94 - 10.9i)T + (-105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-21.5 - 6.99i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-27.3 - 84.0i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (60.6 + 83.4i)T + (-2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 42.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-6.11 - 4.44i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (67.5 - 207. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-251. - 182. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-268. + 195. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 419. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (100. + 137. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-329. - 107. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-30.0 + 41.3i)T + (-6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-52.8 + 17.1i)T + (1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 5.78T + 3.00e5T^{2} \) |
| 71 | \( 1 + (69.3 - 22.5i)T + (2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (170. - 234. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-605. - 196. i)T + (3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (360. + 1.10e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 1.31e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-189. + 582. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.55849161286916433659779169139, −15.62915681842492704145240196559, −14.99385931712383447228294211264, −12.54815534590761984095353356834, −11.72775410020177208681464996971, −10.85631472886570803285443125324, −8.726240789898010649372932459612, −7.15662512792549060479879571745, −6.14181679904339010542330058739, −3.77597883681779662834918237450,
0.74872092527057335419796482555, 4.08186175877099187669032469588, 6.22535341990677789546060922261, 7.54718721635890403825427721670, 9.709699981301694943122131578552, 11.08780518985144746790431474374, 11.78946442380984703164012673014, 12.77530349902044628574773549301, 14.80144591564140971782689261592, 16.19651541648292020255777260973
