L(s) = 1 | + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (−0.0980 − 0.995i)7-s + (0.995 + 0.0980i)8-s + (−0.634 − 0.773i)9-s + (−1.93 + 0.0951i)11-s + (−0.831 + 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (0.997 + 1.66i)22-s + (−0.728 + 1.36i)23-s + (0.956 + 0.290i)25-s + (0.881 + 0.471i)28-s + (−0.217 + 0.197i)29-s + (−0.634 + 0.773i)32-s + ⋯ |
L(s) = 1 | + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (−0.0980 − 0.995i)7-s + (0.995 + 0.0980i)8-s + (−0.634 − 0.773i)9-s + (−1.93 + 0.0951i)11-s + (−0.831 + 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (0.997 + 1.66i)22-s + (−0.728 + 1.36i)23-s + (0.956 + 0.290i)25-s + (0.881 + 0.471i)28-s + (−0.217 + 0.197i)29-s + (−0.634 + 0.773i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1957245563\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1957245563\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.471 + 0.881i)T \) |
| 7 | \( 1 + (0.0980 + 0.995i)T \) |
good | 3 | \( 1 + (0.634 + 0.773i)T^{2} \) |
| 5 | \( 1 + (-0.956 - 0.290i)T^{2} \) |
| 11 | \( 1 + (1.93 - 0.0951i)T + (0.995 - 0.0980i)T^{2} \) |
| 13 | \( 1 + (-0.290 - 0.956i)T^{2} \) |
| 17 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 19 | \( 1 + (-0.881 - 0.471i)T^{2} \) |
| 23 | \( 1 + (0.728 - 1.36i)T + (-0.555 - 0.831i)T^{2} \) |
| 29 | \( 1 + (0.217 - 0.197i)T + (0.0980 - 0.995i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (1.71 + 1.02i)T + (0.471 + 0.881i)T^{2} \) |
| 41 | \( 1 + (-0.831 + 0.555i)T^{2} \) |
| 43 | \( 1 + (1.21 - 0.574i)T + (0.634 - 0.773i)T^{2} \) |
| 47 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 53 | \( 1 + (0.499 + 0.452i)T + (0.0980 + 0.995i)T^{2} \) |
| 59 | \( 1 + (-0.290 + 0.956i)T^{2} \) |
| 61 | \( 1 + (-0.773 + 0.634i)T^{2} \) |
| 67 | \( 1 + (0.609 + 1.70i)T + (-0.773 + 0.634i)T^{2} \) |
| 71 | \( 1 + (0.301 + 0.247i)T + (0.195 + 0.980i)T^{2} \) |
| 73 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 79 | \( 1 + (0.113 + 0.569i)T + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.471 + 0.881i)T^{2} \) |
| 89 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
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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
−9.149559483315402269449384054958, −8.225192652510965548851212470188, −7.64187021336730339124345089561, −6.87961483544832181331403216249, −5.54670155179172277668884291645, −4.77494238206770436262759687650, −3.55978015895964031716844697781, −3.06060042387548082657017742979, −1.75011280375004429612024393860, −0.16029238220348089392427428429,
2.10524724599924165294311447639, 2.94534561530824284198070873759, 4.70856423362891825247711130857, 5.25265101012469532639041256669, 5.88675528091386924780144929810, 6.81509129228228833266725071247, 7.77968902579534365473411477028, 8.521188984166169862626848306568, 8.644430980279100902216905281113, 10.03866308608532268680879917672