L(s) = 1 | + (2.5 + 4.33i)5-s + (−14.9 + 25.9i)7-s + (−7.95 + 13.7i)11-s + (−20.9 − 36.3i)13-s − 36.9·17-s − 10.9·19-s + (57.3 + 99.3i)23-s + (−12.5 + 21.6i)25-s + (−72.7 + 126. i)29-s + (23.2 + 40.2i)31-s − 149.·35-s + 74·37-s + (−167. − 290. i)41-s + (−42.2 + 73.0i)43-s + (−127. + 221. i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.807 + 1.39i)7-s + (−0.218 + 0.377i)11-s + (−0.447 − 0.774i)13-s − 0.526·17-s − 0.131·19-s + (0.519 + 0.900i)23-s + (−0.100 + 0.173i)25-s + (−0.466 + 0.807i)29-s + (0.134 + 0.233i)31-s − 0.722·35-s + 0.328·37-s + (−0.638 − 1.10i)41-s + (−0.149 + 0.259i)43-s + (−0.396 + 0.685i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1552272908\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1552272908\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 7 | \( 1 + (14.9 - 25.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (7.95 - 13.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (20.9 + 36.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 36.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-57.3 - 99.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (72.7 - 126. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-23.2 - 40.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 74T + 5.06e4T^{2} \) |
| 41 | \( 1 + (167. + 290. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (42.2 - 73.0i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (127. - 221. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 399.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (1.28 + 2.21i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (284. - 492. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (384. + 666. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 441.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 966.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (251. - 435. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-635. + 1.10e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 0.240T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-3.35 + 5.81i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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
−9.036103254483313411055913817613, −8.024334859493719443080102697705, −7.14344878684584039546824561887, −6.32834440727651234907312010032, −5.57346628868798261164943666659, −4.87469662850903091398053194819, −3.40096487614082943402396712019, −2.75902254570154864297032359171, −1.81005599507184486496156518408, −0.03927333388567705552260916062,
0.877849678506526531314915767479, 2.19817414366813298802530399118, 3.37725204880363657667892370204, 4.26865044784176658031509555054, 4.96000249205269303861416986305, 6.28308730280209992247996623124, 6.72463597507564743970126935014, 7.62175543168577851093842265426, 8.455834109224853997788446036953, 9.424966519881626039115676744839