L(s) = 1 | + (11.8 + 10.1i)3-s − 32·4-s − 111. i·5-s + 113.·7-s + (37.4 + 240. i)9-s + (−378. − 324. i)12-s + (1.13e3 − 1.32e3i)15-s + 1.02e3·16-s − 683. i·17-s − 3.06e3·19-s + 3.57e3i·20-s + (1.34e3 + 1.14e3i)21-s − 9.32e3·25-s + (−1.99e3 + 3.22e3i)27-s − 3.62e3·28-s − 8.94e3i·29-s + ⋯ |
L(s) = 1 | + (0.759 + 0.650i)3-s − 4-s − 1.99i·5-s + 0.873·7-s + (0.153 + 0.988i)9-s + (−0.759 − 0.650i)12-s + (1.29 − 1.51i)15-s + 16-s − 0.573i·17-s − 1.94·19-s + 1.99i·20-s + (0.663 + 0.567i)21-s − 2.98·25-s + (−0.525 + 0.850i)27-s − 0.873·28-s − 1.97i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.235739929\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.235739929\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-11.8 - 10.1i)T \) |
| 59 | \( 1 + 2.67e4iT \) |
good | 2 | \( 1 + 32T^{2} \) |
| 5 | \( 1 + 111. iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 113.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 + 683. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 3.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.94e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.93e7T^{2} \) |
| 41 | \( 1 - 524. iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.04e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−11.52154894102574582906262426242, −10.01322406985471398090600689297, −9.246747296712957424834576522190, −8.349473905029746918382058117713, −8.124401706294861240995090120632, −5.50996238274063698729212169954, −4.57314034280603070126093001327, −4.15745704638152719725346799770, −1.87304051097798361934430712277, −0.35775265686656443075088849696,
1.79881452584896459338560862693, 3.09290071687403850225322026677, 4.19522742716935936313883447974, 6.06413557138978164724247350829, 7.09452088091934992560493726445, 8.049588177425549433208313755880, 8.931000952357777758696073334470, 10.29495426393076973185579000017, 10.95117470614282094040707181641, 12.31602112473714578560486821695