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
−15.78342256371999102310315179802, −13.75907895767824164724367519809, −12.42860853478052352325687088681, −11.45635886415112434371157645258, −10.39875332445655324708699657774, −9.598464690110522423787721588700, −8.071798025618221242739724879443, −4.67005612424613269613916905744, −3.30898920005320025250843532531, −0.14515877396324156328585141957,
5.29788502166139771433919241820, 6.53042038287895966630924149541, 7.67232409657361899556210538583, 8.662257448563688394314268561958, 10.49257608625194179626167907966, 12.49187294997457263276950754843, 13.69494976779638517364672242600, 15.06993436016694079046216552586, 15.90506317881288574914296818753, 16.94825407771734689743871163229