L(s) = 1 | + (5.54 − 5.54i)3-s + (−21.7 + 21.7i)5-s + 6.62·7-s + 19.6i·9-s + (−90.9 − 90.9i)11-s + (−221. − 221. i)13-s + 240. i·15-s − 132.·17-s + (−402. + 402. i)19-s + (36.6 − 36.6i)21-s − 27.5·23-s − 320. i·25-s + (557. + 557. i)27-s + (−174. − 174. i)29-s + 1.08e3i·31-s + ⋯ |
L(s) = 1 | + (0.615 − 0.615i)3-s + (−0.869 + 0.869i)5-s + 0.135·7-s + 0.242i·9-s + (−0.752 − 0.752i)11-s + (−1.31 − 1.31i)13-s + 1.07i·15-s − 0.458·17-s + (−1.11 + 1.11i)19-s + (0.0831 − 0.0831i)21-s − 0.0519·23-s − 0.512i·25-s + (0.764 + 0.764i)27-s + (−0.207 − 0.207i)29-s + 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0101240 + 0.0769992i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0101240 + 0.0769992i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-5.54 + 5.54i)T - 81iT^{2} \) |
| 5 | \( 1 + (21.7 - 21.7i)T - 625iT^{2} \) |
| 7 | \( 1 - 6.62T + 2.40e3T^{2} \) |
| 11 | \( 1 + (90.9 + 90.9i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (221. + 221. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 132.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (402. - 402. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 27.5T + 2.79e5T^{2} \) |
| 29 | \( 1 + (174. + 174. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 1.08e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (553. - 553. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.80e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-17.8 - 17.8i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.26e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-822. + 822. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (972. + 972. i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (-2.05e3 - 2.05e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (-4.61e3 + 4.61e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 3.10e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 723. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.41e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (161. - 161. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.26e3T + 8.85e7T^{2} \) |
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
−13.06948646855260768563726683057, −12.25109891667794787951933865277, −10.90972556150189613980629029870, −10.27884980989936724283420397785, −8.378363941741067394634602669489, −7.85181044226004210971088427528, −6.89416460520642841276622100713, −5.24398408801333593246759118683, −3.42798132716934368461864979620, −2.34458125383439833380869397453,
0.02770355095177397624396912803, 2.39281634571362825484142718452, 4.25213017562983597458560496115, 4.73165082340308640942486802189, 6.85298120680978566881466648711, 8.037987933457696567090552907535, 9.054469565351955462983637984636, 9.769386343181256234620926553847, 11.22400455264721240669725572400, 12.23164819215256632476865749114