| L(s) = 1 | + (−0.894 + 1.25i)3-s + (2.41 − 0.464i)5-s + (−0.317 − 2.62i)7-s + (0.204 + 0.589i)9-s + (4.00 − 3.15i)11-s + (−1.69 + 1.08i)13-s + (−1.57 + 3.44i)15-s + (−3.47 − 3.31i)17-s + (5.45 − 5.19i)19-s + (3.58 + 1.94i)21-s + (−4.72 + 0.844i)23-s + (0.957 − 0.383i)25-s + (−5.35 − 1.57i)27-s + (8.44 − 2.47i)29-s + (6.28 − 0.599i)31-s + ⋯ |
| L(s) = 1 | + (−0.516 + 0.724i)3-s + (1.07 − 0.207i)5-s + (−0.119 − 0.992i)7-s + (0.0680 + 0.196i)9-s + (1.20 − 0.949i)11-s + (−0.469 + 0.301i)13-s + (−0.405 + 0.889i)15-s + (−0.843 − 0.804i)17-s + (1.25 − 1.19i)19-s + (0.781 + 0.425i)21-s + (−0.984 + 0.176i)23-s + (0.191 − 0.0766i)25-s + (−1.03 − 0.302i)27-s + (1.56 − 0.460i)29-s + (1.12 − 0.107i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54968 - 0.134936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54968 - 0.134936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.317 + 2.62i)T \) |
| 23 | \( 1 + (4.72 - 0.844i)T \) |
| good | 3 | \( 1 + (0.894 - 1.25i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-2.41 + 0.464i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-4.00 + 3.15i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.69 - 1.08i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.47 + 3.31i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.45 + 5.19i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-8.44 + 2.47i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-6.28 + 0.599i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-2.15 - 6.23i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-6.09 - 7.03i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.06 - 4.51i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-2.29 + 3.98i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.179 - 3.76i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (7.29 + 3.76i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-3.74 - 5.25i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (1.57 - 0.629i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.475 - 3.30i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.08 + 8.57i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.123 - 2.59i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-4.62 + 5.34i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.31 - 0.412i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-2.82 - 3.26i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.40961651179714637604900311239, −9.667454318564677125567931048835, −9.255013561352318575716674051308, −7.913525774356973379054025855310, −6.72435137829345637028783604034, −6.06461040544516965067166182090, −4.86530130548020383967976517621, −4.28345348924769183722698684768, −2.77718204218529148026480379981, −1.04605084411455398476360747183,
1.48313240401060519611805017107, 2.41256409242000004707785472958, 4.05406928584980872115668553221, 5.47918586775250785544036614880, 6.19176306037415744819436347941, 6.70941339412642520843645051318, 7.85303237328311933584391927157, 9.075701917401763073316129293363, 9.684127092386704846734406596371, 10.43823530038718431555062118325
