| L(s) = 1 | + (1.32 − 0.505i)2-s + (0.0822 + 0.0877i)3-s + (1.48 − 1.33i)4-s + (0.463 + 1.02i)5-s + (0.153 + 0.0743i)6-s + (1.41 − 2.23i)7-s + (1.28 − 2.51i)8-s + (0.195 − 2.97i)9-s + (1.13 + 1.11i)10-s + (−3.23 − 5.20i)11-s + (0.239 + 0.0207i)12-s + (0.512 + 5.20i)13-s + (0.740 − 3.66i)14-s + (−0.0517 + 0.124i)15-s + (0.429 − 3.97i)16-s + (−4.39 − 0.578i)17-s + ⋯ |
| L(s) = 1 | + (0.933 − 0.357i)2-s + (0.0474 + 0.0506i)3-s + (0.744 − 0.668i)4-s + (0.207 + 0.457i)5-s + (0.0624 + 0.0303i)6-s + (0.535 − 0.844i)7-s + (0.455 − 0.890i)8-s + (0.0650 − 0.993i)9-s + (0.357 + 0.353i)10-s + (−0.975 − 1.56i)11-s + (0.0691 + 0.00600i)12-s + (0.142 + 1.44i)13-s + (0.197 − 0.980i)14-s + (−0.0133 + 0.0322i)15-s + (0.107 − 0.994i)16-s + (−1.06 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.26588 - 1.83781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26588 - 1.83781i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 + 0.505i)T \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
| good | 3 | \( 1 + (-0.0822 - 0.0877i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.463 - 1.02i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (3.23 + 5.20i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (-0.512 - 5.20i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (4.39 + 0.578i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (0.0934 - 0.565i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-2.46 - 2.80i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-2.97 - 5.56i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (-0.948 + 0.254i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (1.73 - 4.60i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-1.20 - 6.06i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-5.01 + 1.52i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-3.60 + 4.70i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (1.25 + 2.01i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (5.47 - 7.63i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (-5.90 + 1.37i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-8.25 - 8.80i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-5.93 - 8.87i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-11.4 + 5.65i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 2.35i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-4.09 + 4.98i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-0.394 - 1.16i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (13.1 - 13.1i)T - 97iT^{2} \) |
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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
−10.25561603466878342427379967401, −9.209877721596439504950639349050, −8.260915312055529874869576713188, −6.86502302069702595255661599860, −6.61052356454933564152002215043, −5.44603413704593396139109730554, −4.42833468253181518191508241295, −3.57149001788648853876935563045, −2.59448876941490408781988033389, −1.07762860196113553064762556110,
2.09691447302730167853846407060, 2.66494711934726099450898819025, 4.42599807355584402449100170910, 5.06523431441972039422966069984, 5.58269867102477356611255263901, 6.84639310802944489690216165582, 7.83744772919366765425544990847, 8.222526354209419664768765773983, 9.374238302293900489126556633490, 10.68655092009624215386824237721
