| L(s) = 1 | + (−2.04 + 1.27i)2-s + (2.34 + 0.671i)3-s + (1.66 − 3.40i)4-s + (−1.72 + 1.42i)5-s + (−5.63 + 1.61i)6-s + (−2.03 − 3.52i)7-s + (0.449 + 4.27i)8-s + (2.48 + 1.55i)9-s + (1.69 − 5.10i)10-s + (−3.19 − 3.54i)11-s + (6.17 − 6.85i)12-s + (0.0693 − 1.98i)13-s + (8.63 + 4.59i)14-s + (−4.98 + 2.18i)15-s + (−1.70 − 2.18i)16-s + (3.81 + 0.266i)17-s + ⋯ |
| L(s) = 1 | + (−1.44 + 0.901i)2-s + (1.35 + 0.387i)3-s + (0.830 − 1.70i)4-s + (−0.769 + 0.638i)5-s + (−2.29 + 0.659i)6-s + (−0.768 − 1.33i)7-s + (0.159 + 1.51i)8-s + (0.827 + 0.517i)9-s + (0.535 − 1.61i)10-s + (−0.963 − 1.06i)11-s + (1.78 − 1.97i)12-s + (0.0192 − 0.550i)13-s + (2.30 + 1.22i)14-s + (−1.28 + 0.563i)15-s + (−0.427 − 0.547i)16-s + (0.925 + 0.0646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.647765 - 0.148229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647765 - 0.148229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.72 - 1.42i)T \) |
| 19 | \( 1 + (-2.70 + 3.41i)T \) |
| good | 2 | \( 1 + (2.04 - 1.27i)T + (0.876 - 1.79i)T^{2} \) |
| 3 | \( 1 + (-2.34 - 0.671i)T + (2.54 + 1.58i)T^{2} \) |
| 7 | \( 1 + (2.03 + 3.52i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.19 + 3.54i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0693 + 1.98i)T + (-12.9 - 0.906i)T^{2} \) |
| 17 | \( 1 + (-3.81 - 0.266i)T + (16.8 + 2.36i)T^{2} \) |
| 23 | \( 1 + (-9.21 - 1.29i)T + (22.1 + 6.33i)T^{2} \) |
| 29 | \( 1 + (8.12 - 0.567i)T + (28.7 - 4.03i)T^{2} \) |
| 31 | \( 1 + (6.73 + 2.99i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.98 - 6.10i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.23 + 2.85i)T + (-9.91 + 39.7i)T^{2} \) |
| 43 | \( 1 + (-1.76 + 9.98i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.90 + 0.272i)T + (46.5 - 6.54i)T^{2} \) |
| 53 | \( 1 + (1.22 - 2.50i)T + (-32.6 - 41.7i)T^{2} \) |
| 59 | \( 1 + (-1.25 + 3.11i)T + (-42.4 - 40.9i)T^{2} \) |
| 61 | \( 1 + (2.93 + 0.412i)T + (58.6 + 16.8i)T^{2} \) |
| 67 | \( 1 + (-0.518 + 2.08i)T + (-59.1 - 31.4i)T^{2} \) |
| 71 | \( 1 + (3.73 - 3.61i)T + (2.47 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-0.227 - 6.52i)T + (-72.8 + 5.09i)T^{2} \) |
| 79 | \( 1 + (-2.33 - 0.670i)T + (66.9 + 41.8i)T^{2} \) |
| 83 | \( 1 + (1.60 + 0.713i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.20 + 2.82i)T + (-21.5 - 86.3i)T^{2} \) |
| 97 | \( 1 + (0.561 + 2.25i)T + (-85.6 + 45.5i)T^{2} \) |
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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
−10.61741236326844755455739601896, −9.790787589174091859385211474813, −9.029708200670617574431377548371, −8.162413719996686737020448306144, −7.40776014654275004139016218259, −7.12044872398758951001579335050, −5.58189655785396956926247976674, −3.63520371568314988652688139925, −3.04230265676980159732381988158, −0.55558278727762373083123904187,
1.65034901120863431367118221988, 2.71150077684888130100566403504, 3.47347636834110386553163260251, 5.31812438772620239888637405057, 7.36225223009432680817612812939, 7.68725109093277311275689539012, 8.688736446495091023575366946188, 9.255584556094436911489897046938, 9.648917451913506313883748423219, 10.97331621604836050038542384062
