L(s) = 1 | + (1.22 − 0.707i)2-s + (−1.44 + 2.63i)3-s + (0.999 − 1.73i)4-s + (1.93 − 1.11i)5-s + (0.0971 + 4.24i)6-s + (1.98 − 6.71i)7-s − 2.82i·8-s + (−4.85 − 7.58i)9-s + (1.58 − 2.73i)10-s + (11.3 + 6.55i)11-s + (3.11 + 5.12i)12-s + 20.0·13-s + (−2.31 − 9.62i)14-s + (0.153 + 6.70i)15-s + (−2.00 − 3.46i)16-s + (7.44 + 4.29i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.480 + 0.877i)3-s + (0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.0161 + 0.706i)6-s + (0.283 − 0.958i)7-s − 0.353i·8-s + (−0.539 − 0.842i)9-s + (0.158 − 0.273i)10-s + (1.03 + 0.595i)11-s + (0.259 + 0.427i)12-s + 1.53·13-s + (−0.165 − 0.687i)14-s + (0.0102 + 0.447i)15-s + (−0.125 − 0.216i)16-s + (0.437 + 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.10713 - 0.348953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10713 - 0.348953i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (1.44 - 2.63i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (-1.98 + 6.71i)T \) |
good | 11 | \( 1 + (-11.3 - 6.55i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 20.0T + 169T^{2} \) |
| 17 | \( 1 + (-7.44 - 4.29i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.25 - 5.63i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (33.1 - 19.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 23.6iT - 841T^{2} \) |
| 31 | \( 1 + (-22.0 + 38.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (9.37 + 16.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 14.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (69.0 - 39.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (22.0 + 12.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-87.3 - 50.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (24.7 + 42.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (29.3 - 50.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 82.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (34.8 - 60.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (0.932 + 1.61i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 123. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-53.2 + 30.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 113.T + 9.40e3T^{2} \) |
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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
−11.79867050793923105366246830386, −11.30056003201927750120394935218, −10.13189951191609096617846310666, −9.633406753285726389016728416370, −8.175483833637924358363591632744, −6.51419570181646003081438574298, −5.70469720203800890146836107564, −4.28456089871689116751288859427, −3.72555334333461419258990105561, −1.35101416763504484285953705359,
1.63422776601355304465235360850, 3.27788053450025397813089610729, 5.09421996980146132048759835163, 6.16531901778672807625040504219, 6.58598334378988785291140788807, 8.162608775805713319926301550516, 8.839698085016177475619308279621, 10.57566599361259592982667236551, 11.65483155719356016689107578025, 12.08733266723382647001700108362