| L(s) = 1 | + (−0.729 + 0.529i)2-s + (−0.361 + 0.160i)3-s + (−0.367 + 1.12i)4-s + (−0.5 − 0.866i)5-s + (0.178 − 0.308i)6-s + (2.69 + 2.99i)7-s + (−0.887 − 2.73i)8-s + (−1.90 + 2.11i)9-s + (0.823 + 0.366i)10-s + (−4.89 + 1.04i)11-s + (−0.0490 − 0.466i)12-s + (−0.470 + 4.47i)13-s + (−3.54 − 0.754i)14-s + (0.319 + 0.232i)15-s + (0.172 + 0.125i)16-s + (5.25 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.515 + 0.374i)2-s + (−0.208 + 0.0928i)3-s + (−0.183 + 0.564i)4-s + (−0.223 − 0.387i)5-s + (0.0727 − 0.125i)6-s + (1.01 + 1.13i)7-s + (−0.313 − 0.966i)8-s + (−0.634 + 0.704i)9-s + (0.260 + 0.115i)10-s + (−1.47 + 0.313i)11-s + (−0.0141 − 0.134i)12-s + (−0.130 + 1.24i)13-s + (−0.948 − 0.201i)14-s + (0.0825 + 0.0599i)15-s + (0.0432 + 0.0313i)16-s + (1.27 + 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.323912 + 0.579654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323912 + 0.579654i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-4.42 - 3.37i)T \) |
| good | 2 | \( 1 + (0.729 - 0.529i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.361 - 0.160i)T + (2.00 - 2.22i)T^{2} \) |
| 7 | \( 1 + (-2.69 - 2.99i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (4.89 - 1.04i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.470 - 4.47i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-5.25 - 1.11i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.449 + 4.28i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.647 - 1.99i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.12 + 2.99i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.763 + 1.32i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 0.481i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (1.00 + 9.55i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-1.80 - 1.31i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.77 + 4.18i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (8.62 - 3.83i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 8.99T + 61T^{2} \) |
| 67 | \( 1 + (-7.23 - 12.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.50 - 10.5i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (0.175 - 0.0372i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (7.30 + 1.55i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (0.954 + 0.425i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.47 + 10.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.13 + 3.49i)T + (-78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.20052941169415586528786569205, −12.09579545715557358003490972623, −11.51668900913400614306941919264, −10.12869742667348579649450722551, −8.787510948012905382371945866001, −8.264971198419405605364522671989, −7.32749616104960507930230950732, −5.54357045213960847053220376883, −4.64243593559869062062547636473, −2.55121922372272549819526925665,
0.808080972784207047564424827603, 3.03471379883311419646163722499, 4.95701777525137146124448196540, 5.95575133797321120554209549559, 7.75294883072182583119749661407, 8.245462996105695288246392843389, 9.991170322218985966082455518687, 10.53801686847827104746945209952, 11.26464852189172495055264201545, 12.41540956297584703974557793362
