| L(s) = 1 | + (1.11 − 0.869i)2-s + (0.793 + 1.53i)3-s + (0.488 − 1.93i)4-s + (1.27 + 1.98i)5-s + (2.22 + 1.02i)6-s + (0.413 + 0.0594i)7-s + (−1.14 − 2.58i)8-s + (−1.73 + 2.44i)9-s + (3.15 + 1.10i)10-s + (−1.97 − 4.33i)11-s + (3.37 − 0.788i)12-s + (0.584 + 4.06i)13-s + (0.512 − 0.293i)14-s + (−2.04 + 3.54i)15-s + (−3.52 − 1.89i)16-s + (0.621 + 0.538i)17-s + ⋯ |
| L(s) = 1 | + (0.788 − 0.614i)2-s + (0.458 + 0.888i)3-s + (0.244 − 0.969i)4-s + (0.570 + 0.888i)5-s + (0.907 + 0.419i)6-s + (0.156 + 0.0224i)7-s + (−0.403 − 0.914i)8-s + (−0.579 + 0.814i)9-s + (0.996 + 0.349i)10-s + (−0.596 − 1.30i)11-s + (0.973 − 0.227i)12-s + (0.162 + 1.12i)13-s + (0.137 − 0.0783i)14-s + (−0.527 + 0.914i)15-s + (−0.880 − 0.473i)16-s + (0.150 + 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.26601 + 0.0154767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26601 + 0.0154767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.11 + 0.869i)T \) |
| 3 | \( 1 + (-0.793 - 1.53i)T \) |
| 23 | \( 1 + (-1.79 + 4.44i)T \) |
| good | 5 | \( 1 + (-1.27 - 1.98i)T + (-2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.413 - 0.0594i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.97 + 4.33i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.584 - 4.06i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.621 - 0.538i)T + (2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.599 + 0.519i)T + (2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (6.93 + 6.00i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.502 + 1.71i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-6.75 - 4.34i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.51 + 2.35i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (2.85 - 9.72i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + (-0.00652 - 0.000938i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.17 - 8.18i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-12.6 + 3.72i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.07 + 1.40i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.499 + 1.09i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.91 - 3.35i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (8.39 - 1.20i)T + (75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-8.80 - 5.66i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (1.69 - 5.78i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (11.5 - 7.40i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.39804428846519479624425161385, −11.14611480635699140180755592570, −10.13629940946106681488896468451, −9.427003854042895937582324617456, −8.210703528100676359150229778481, −6.55503964820376488260935590607, −5.67026526518204467816666080782, −4.45189482103240854518190839690, −3.27906336767037594223344813054, −2.33142551168128733026626646702,
1.87348557873881572002134495932, 3.35332396777201884158321605150, 5.02998868025787718627934056997, 5.68288656952058488877622944173, 7.09352828860415288223383348960, 7.74838133010265208564218454502, 8.718206119135675036705244390695, 9.727939449003149407998883705548, 11.30881608108958806690040981265, 12.49382135950575379763020224643
