| L(s) = 1 | + (0.909 − 1.08i)2-s + (1.54 − 2.56i)3-s + (−0.347 − 1.96i)4-s + (−1.07 + 2.96i)5-s + (−1.37 − 4.01i)6-s + (−0.250 + 1.42i)7-s + (−2.44 − 1.41i)8-s + (−4.20 − 7.95i)9-s + (2.23 + 3.86i)10-s + (6.39 + 17.5i)11-s + (−5.59 − 2.15i)12-s + (10.5 − 8.88i)13-s + (1.31 + 1.56i)14-s + (5.94 + 7.36i)15-s + (−3.75 + 1.36i)16-s + (−16.3 + 9.44i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.516 − 0.856i)3-s + (−0.0868 − 0.492i)4-s + (−0.215 + 0.592i)5-s + (−0.229 − 0.668i)6-s + (−0.0357 + 0.202i)7-s + (−0.306 − 0.176i)8-s + (−0.466 − 0.884i)9-s + (0.223 + 0.386i)10-s + (0.581 + 1.59i)11-s + (−0.466 − 0.179i)12-s + (0.814 − 0.683i)13-s + (0.0936 + 0.111i)14-s + (0.396 + 0.490i)15-s + (−0.234 + 0.0855i)16-s + (−0.961 + 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29376 - 0.849561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29376 - 0.849561i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 + 1.08i)T \) |
| 3 | \( 1 + (-1.54 + 2.56i)T \) |
| good | 5 | \( 1 + (1.07 - 2.96i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (0.250 - 1.42i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-6.39 - 17.5i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-10.5 + 8.88i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (16.3 - 9.44i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.14 - 1.98i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (26.5 - 4.68i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-17.2 + 20.5i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (4.55 + 25.8i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (33.8 + 58.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (13.0 + 15.5i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (32.2 - 11.7i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-46.4 - 8.19i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + 49.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (13.0 - 35.9i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (9.55 - 54.2i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (95.2 - 79.9i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-10.4 + 6.04i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-37.3 + 64.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-74.8 - 62.7i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-81.1 + 96.6i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-9.23 - 5.33i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-145. + 53.0i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−14.80493152420548777928989870828, −13.64901859511739893711104592282, −12.64936688495860256201038711321, −11.76204231759163151584847428070, −10.38337485194511327614197068157, −8.942280349696096109327819314063, −7.40549408252328593977674806805, −6.16748195500619952805040235798, −3.89381276017032085547010350794, −2.11540003380967289045356349542,
3.52243750840087464983666773797, 4.78639621197895058270507874891, 6.42521568881088734539860975734, 8.419559879556092681782845778218, 8.936488750296122830231855640717, 10.73322793784886599874390618335, 11.90882090216400100612005541930, 13.70456419201752749617229554513, 13.97180324320415481633378282619, 15.51237182317353032532532851263
