L(s) = 1 | + (−1.66 + 5.13i)2-s + (−2.33 − 4.64i)3-s + (−17.1 − 12.4i)4-s + (−7.03 + 2.28i)5-s + (27.7 − 4.22i)6-s + (−8.32 + 11.4i)7-s + (57.6 − 41.8i)8-s + (−16.1 + 21.6i)9-s − 39.9i·10-s + (−36.4 − 1.15i)11-s + (−17.8 + 108. i)12-s + (−5.34 − 1.73i)13-s + (−44.9 − 61.9i)14-s + (26.9 + 27.3i)15-s + (66.6 + 204. i)16-s + (16.4 + 50.7i)17-s + ⋯ |
L(s) = 1 | + (−0.590 + 1.81i)2-s + (−0.448 − 0.893i)3-s + (−2.14 − 1.55i)4-s + (−0.628 + 0.204i)5-s + (1.88 − 0.287i)6-s + (−0.449 + 0.618i)7-s + (2.54 − 1.85i)8-s + (−0.597 + 0.801i)9-s − 1.26i·10-s + (−0.999 − 0.0316i)11-s + (−0.429 + 2.61i)12-s + (−0.113 − 0.0370i)13-s + (−0.858 − 1.18i)14-s + (0.464 + 0.470i)15-s + (1.04 + 3.20i)16-s + (0.235 + 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0635300 - 0.110967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0635300 - 0.110967i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.33 + 4.64i)T \) |
| 11 | \( 1 + (36.4 + 1.15i)T \) |
good | 2 | \( 1 + (1.66 - 5.13i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (7.03 - 2.28i)T + (101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (8.32 - 11.4i)T + (-105. - 326. i)T^{2} \) |
| 13 | \( 1 + (5.34 + 1.73i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-16.4 - 50.7i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (20.6 + 28.4i)T + (-2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 64.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-113. - 82.7i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (9.10 - 28.0i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (171. + 124. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (213. - 154. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 385. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (133. + 184. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (185. + 60.3i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (277. - 381. i)T + (-6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (616. - 200. i)T + (1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (605. - 196. i)T + (2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-110. + 152. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-816. - 265. i)T + (3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-67.7 - 208. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 217. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (537. - 1.65e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.94825407771734689743871163229, −15.90506317881288574914296818753, −15.06993436016694079046216552586, −13.69494976779638517364672242600, −12.49187294997457263276950754843, −10.49257608625194179626167907966, −8.662257448563688394314268561958, −7.67232409657361899556210538583, −6.53042038287895966630924149541, −5.29788502166139771433919241820,
0.14515877396324156328585141957, 3.30898920005320025250843532531, 4.67005612424613269613916905744, 8.071798025618221242739724879443, 9.598464690110522423787721588700, 10.39875332445655324708699657774, 11.45635886415112434371157645258, 12.42860853478052352325687088681, 13.75907895767824164724367519809, 15.78342256371999102310315179802