| L(s) = 1 | + (−2.25 + 0.259i)2-s + (0.692 − 0.503i)3-s + (3.08 − 0.718i)4-s + (0.564 + 0.825i)5-s + (−1.43 + 1.31i)6-s + (−1.93 + 1.98i)7-s + (−2.50 + 0.895i)8-s + (−0.700 + 2.15i)9-s + (−1.48 − 1.71i)10-s + (0.390 + 3.29i)11-s + (1.77 − 2.05i)12-s + (−1.82 − 3.02i)13-s + (3.85 − 4.99i)14-s + (0.806 + 0.287i)15-s + (−0.255 + 0.125i)16-s + (−2.28 − 1.86i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 + 0.183i)2-s + (0.399 − 0.290i)3-s + (1.54 − 0.359i)4-s + (0.252 + 0.369i)5-s + (−0.585 + 0.537i)6-s + (−0.730 + 0.752i)7-s + (−0.887 + 0.316i)8-s + (−0.233 + 0.718i)9-s + (−0.470 − 0.543i)10-s + (0.117 + 0.993i)11-s + (0.513 − 0.592i)12-s + (−0.505 − 0.838i)13-s + (1.02 − 1.33i)14-s + (0.208 + 0.0742i)15-s + (−0.0639 + 0.0314i)16-s + (−0.553 − 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00740916 - 0.139194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00740916 - 0.139194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.564 - 0.825i)T \) |
| 11 | \( 1 + (-0.390 - 3.29i)T \) |
| good | 2 | \( 1 + (2.25 - 0.259i)T + (1.94 - 0.452i)T^{2} \) |
| 3 | \( 1 + (-0.692 + 0.503i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (1.93 - 1.98i)T + (-0.199 - 6.99i)T^{2} \) |
| 13 | \( 1 + (1.82 + 3.02i)T + (-6.06 + 11.4i)T^{2} \) |
| 17 | \( 1 + (2.28 + 1.86i)T + (3.37 + 16.6i)T^{2} \) |
| 19 | \( 1 + (5.03 - 0.288i)T + (18.8 - 2.16i)T^{2} \) |
| 23 | \( 1 + (1.10 + 7.69i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + (2.18 + 5.62i)T + (-21.3 + 19.6i)T^{2} \) |
| 31 | \( 1 + (-0.564 + 0.238i)T + (21.6 - 22.2i)T^{2} \) |
| 37 | \( 1 + (-0.609 - 7.09i)T + (-36.4 + 6.30i)T^{2} \) |
| 41 | \( 1 + (-3.45 - 1.94i)T + (21.1 + 35.1i)T^{2} \) |
| 43 | \( 1 + (5.62 + 1.65i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (1.43 + 7.08i)T + (-43.2 + 18.2i)T^{2} \) |
| 53 | \( 1 + (5.47 + 2.69i)T + (32.3 + 41.9i)T^{2} \) |
| 59 | \( 1 + (1.34 - 0.757i)T + (30.4 - 50.5i)T^{2} \) |
| 61 | \( 1 + (6.51 + 0.747i)T + (59.4 + 13.8i)T^{2} \) |
| 67 | \( 1 + (3.09 + 6.77i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.71 - 6.50i)T + (-61.8 - 34.9i)T^{2} \) |
| 73 | \( 1 + (5.05 - 9.57i)T + (-41.2 - 60.2i)T^{2} \) |
| 79 | \( 1 + (0.0128 - 0.449i)T + (-78.8 - 4.51i)T^{2} \) |
| 83 | \( 1 + (3.33 - 0.576i)T + (78.1 - 27.8i)T^{2} \) |
| 89 | \( 1 + (-8.50 + 5.46i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.23 - 4.73i)T + (-35.1 - 90.3i)T^{2} \) |
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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
−10.60184304265797304912400590357, −10.09736886684513063492700646860, −9.314418431356855718693904815552, −8.479783782307716321110288300226, −7.83972023460715470858996370320, −6.87305605165286113593292001652, −6.19712901767187509006906123024, −4.70785302138849345802911881352, −2.65854927112102208838945133585, −2.10498837217638054668256029971,
0.11584940846887323955568221287, 1.66360708250727444008558415816, 3.16701557009668758897776574874, 4.25711990612935092360855926495, 6.04320107401711904145937969451, 6.85392148647477483531801398870, 7.81912416616576614490124869984, 8.942261310485150993708114445742, 9.116768540123507470202810618141, 9.951242412389317542024381177517
