L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.52 + 2.16i)7-s + 0.999i·8-s + (−4.29 − 2.48i)11-s − 5.49i·13-s + (0.243 − 2.63i)14-s + (−0.5 − 0.866i)16-s + (−1.53 + 2.66i)17-s + (−2.68 + 1.55i)19-s + 4.96·22-s + (−5.34 + 3.08i)23-s + (2.74 + 4.75i)26-s + (1.10 + 2.40i)28-s − 6.67i·29-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.577 + 0.816i)7-s + 0.353i·8-s + (−1.29 − 0.748i)11-s − 1.52i·13-s + (0.0649 − 0.704i)14-s + (−0.125 − 0.216i)16-s + (−0.372 + 0.645i)17-s + (−0.616 + 0.355i)19-s + 1.05·22-s + (−1.11 + 0.643i)23-s + (0.538 + 0.933i)26-s + (0.209 + 0.454i)28-s − 1.24i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8100409405\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8100409405\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.52 - 2.16i)T \) |
good | 11 | \( 1 + (4.29 + 2.48i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.49iT - 13T^{2} \) |
| 17 | \( 1 + (1.53 - 2.66i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.68 - 1.55i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.34 - 3.08i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.67iT - 29T^{2} \) |
| 31 | \( 1 + (1.01 + 0.586i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.35 - 9.27i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + (-2.07 - 3.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.85 - 2.22i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.00 - 5.20i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.05 + 5.22i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.97 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.973iT - 71T^{2} \) |
| 73 | \( 1 + (-14.4 - 8.34i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.12 - 3.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + (7.38 + 12.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.41iT - 97T^{2} \) |
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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
−8.493822222807929673564238223640, −8.096108252378940288849777944698, −7.59656842577801650798922557906, −6.23203858200623784247298053003, −5.92218604953307586951356505778, −5.30277283962747458372217062357, −4.02011692696885805924957976260, −2.88539458275294244462986318555, −2.27946686834360086229615661951, −0.61981748360085973359388694951,
0.51716700505482136521624582205, 2.05894314512069956037421169378, 2.63550882182251068324481616360, 4.01448444151921591806151365425, 4.41531768986557926644222452364, 5.60049630871161184538034453371, 6.72247760588360328090367553500, 7.12118648484203403964700050551, 7.81511493612297537042135725018, 8.745998110460983350762250032828