L(s) = 1 | + (0.254 + 2.22i)5-s + 2.64i·7-s + 1.51i·11-s + 3.87·13-s + 3.31i·17-s − 7.08i·19-s + 4.82i·23-s + (−4.87 + 1.12i)25-s + 2.18i·29-s + 7.36·31-s + (−5.87 + 0.671i)35-s − 7.87·37-s − 8.72·41-s − 1.01·43-s + 7.08i·47-s + ⋯ |
L(s) = 1 | + (0.113 + 0.993i)5-s + 0.998i·7-s + 0.456i·11-s + 1.07·13-s + 0.803i·17-s − 1.62i·19-s + 1.00i·23-s + (−0.974 + 0.225i)25-s + 0.405i·29-s + 1.32·31-s + (−0.992 + 0.113i)35-s − 1.29·37-s − 1.36·41-s − 0.155·43-s + 1.03i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.535459161\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535459161\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.254 - 2.22i)T \) |
good | 7 | \( 1 - 2.64iT - 7T^{2} \) |
| 11 | \( 1 - 1.51iT - 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 7.08iT - 19T^{2} \) |
| 23 | \( 1 - 4.82iT - 23T^{2} \) |
| 29 | \( 1 - 2.18iT - 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 - 7.08iT - 47T^{2} \) |
| 53 | \( 1 + 4.50T + 53T^{2} \) |
| 59 | \( 1 + 6.79iT - 59T^{2} \) |
| 61 | \( 1 - 3.60iT - 61T^{2} \) |
| 67 | \( 1 - 1.01T + 67T^{2} \) |
| 71 | \( 1 + 6.72T + 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−9.810226818428075261460677532192, −8.930991393144291870459046366458, −8.314328801644740341250531006040, −7.22120526023901260487492168697, −6.51786876688558097952813339302, −5.80530347700276820238898848802, −4.83587466011782058375013325307, −3.58491394484793003804569950634, −2.77057768298575599662989351449, −1.68848372232822419363629281888,
0.63387276770046806625883227740, 1.69220754793623285507704848024, 3.35456174967073238826948458896, 4.14661494412211771573830890763, 5.02677295304508982351845960538, 5.98571442105713457459918030359, 6.75786698426310170069891003407, 7.911177608819110983359297392566, 8.393844923273724910564295440307, 9.180513667354531415699426001228