| L(s) = 1 | + (0.872 + 3.82i)2-s + (123. + 114. i)3-s + (447. − 215. i)4-s + (1.67e3 − 252. i)5-s + (−329. + 571. i)6-s + (5.96e3 + 1.03e4i)7-s + (2.46e3 + 3.09e3i)8-s + (643. + 8.58e3i)9-s + (2.42e3 + 6.17e3i)10-s + (−1.55e4 − 7.48e3i)11-s + (7.98e4 + 2.46e4i)12-s + (6.05e4 − 1.54e5i)13-s + (−3.42e4 + 3.18e4i)14-s + (2.35e5 + 1.60e5i)15-s + (1.48e5 − 1.86e5i)16-s + (−5.78e5 − 8.72e4i)17-s + ⋯ |
| L(s) = 1 | + (0.0385 + 0.169i)2-s + (0.878 + 0.815i)3-s + (0.873 − 0.420i)4-s + (1.19 − 0.180i)5-s + (−0.103 + 0.179i)6-s + (0.938 + 1.62i)7-s + (0.212 + 0.267i)8-s + (0.0326 + 0.435i)9-s + (0.0766 + 0.195i)10-s + (−0.319 − 0.154i)11-s + (1.11 + 0.342i)12-s + (0.587 − 1.49i)13-s + (−0.238 + 0.221i)14-s + (1.19 + 0.817i)15-s + (0.567 − 0.712i)16-s + (−1.68 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.80930 + 1.70607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.80930 + 1.70607i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.83e7 + 1.28e7i)T \) |
| good | 2 | \( 1 + (-0.872 - 3.82i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-123. - 114. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-1.67e3 + 252. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (-5.96e3 - 1.03e4i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (1.55e4 + 7.48e3i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-6.05e4 + 1.54e5i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (5.78e5 + 8.72e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-9.84e3 + 1.31e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (1.13e6 - 7.71e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (4.00e5 - 3.71e5i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (3.38e6 + 1.04e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (3.40e6 - 5.89e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-4.05e6 - 1.77e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-1.21e7 + 5.84e6i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-3.55e6 - 9.05e6i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (2.83e7 - 3.55e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-1.51e8 + 4.67e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-5.09e6 + 6.80e7i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (1.91e8 + 1.30e8i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (1.20e8 - 3.07e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (2.16e7 + 3.75e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (4.30e8 + 3.99e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-3.54e8 - 3.29e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (3.01e8 + 1.45e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.47041970224495779622024915806, −13.22358774537857229266044439753, −11.55001535567586851616681691490, −10.34905952687483368870428590883, −9.174440766550585996706891562550, −8.233010170258072399608368633619, −6.03570336280622383105220964443, −5.18565686368155827392472803412, −2.77000801230898065326946242939, −1.88794134326109782719631004828,
1.66015555207506046616425614513, 2.11382561351879041278597902591, 4.10420572843090480422070725339, 6.56279058250769563354029171378, 7.35282447083124873131987827112, 8.612894020769718819500579546546, 10.39212335013054882659682388059, 11.29781084476776817369221426858, 13.05030660471221384082060163325, 13.76922499391424516696119294961
