| L(s) = 1 | + (−0.177 − 0.211i)2-s + (1.66 − 0.483i)3-s + (0.334 − 1.89i)4-s + (−0.398 − 0.266i)6-s + (1.66 − 0.293i)7-s + (−0.939 + 0.542i)8-s + (2.53 − 1.60i)9-s + (−4.65 − 1.69i)11-s + (−0.359 − 3.31i)12-s + (2.40 − 2.87i)13-s + (−0.358 − 0.300i)14-s + (−3.33 − 1.21i)16-s + (−5.46 − 3.15i)17-s + (−0.791 − 0.250i)18-s + (3.42 + 5.93i)19-s + ⋯ |
| L(s) = 1 | + (−0.125 − 0.149i)2-s + (0.960 − 0.279i)3-s + (0.167 − 0.947i)4-s + (−0.162 − 0.108i)6-s + (0.628 − 0.110i)7-s + (−0.332 + 0.191i)8-s + (0.844 − 0.535i)9-s + (−1.40 − 0.511i)11-s + (−0.103 − 0.956i)12-s + (0.668 − 0.796i)13-s + (−0.0956 − 0.0802i)14-s + (−0.833 − 0.303i)16-s + (−1.32 − 0.765i)17-s + (−0.186 − 0.0591i)18-s + (0.786 + 1.36i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34726 - 1.46062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34726 - 1.46062i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.66 + 0.483i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.177 + 0.211i)T + (-0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.66 + 0.293i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.65 + 1.69i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 2.87i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.46 + 3.15i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.42 - 5.93i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.37 - 1.12i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.115 + 0.0967i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.06 - 6.03i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.13 - 1.23i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.02 + 2.53i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.39 + 6.57i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.10 + 1.07i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.373iT - 53T^{2} \) |
| 59 | \( 1 + (0.342 - 0.124i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 6.08i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.398 + 0.475i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.39 - 5.88i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.94 + 2.27i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.4 + 10.4i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.18 - 8.56i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.58 + 7.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.70 - 12.9i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.46069481542895841614108595876, −9.318073735268392771598439194967, −8.584318074969922303430166026088, −7.77475296814140187389189977049, −6.89238284870501314744783164800, −5.65218183609776676406919579237, −4.89048082611304069932462450460, −3.33626130883610173541861019211, −2.32536968069300261065786520783, −1.04099859659973540377032032249,
2.12446312785160304776157190694, 2.94401171080427518791022303872, 4.22579765067713639864129000000, 4.92348019135125989846672587270, 6.64300111114736145325463860800, 7.45085432099554891684851298726, 8.165861717412316391502037132519, 8.877190783882189297694125046579, 9.533619782395055695238011114305, 10.97346870881690583027771624924
