| L(s) = 1 | + (1.03 + 0.961i)2-s + (0.381 + 0.606i)3-s + (0.151 + 1.99i)4-s + (−0.388 − 0.309i)5-s + (−0.187 + 0.996i)6-s + (−1.45 − 0.331i)7-s + (−1.76 + 2.21i)8-s + (1.07 − 2.24i)9-s + (−0.105 − 0.694i)10-s + (1.07 − 3.08i)11-s + (−1.15 + 0.852i)12-s + (0.610 + 1.26i)13-s + (−1.18 − 1.73i)14-s + (0.0398 − 0.353i)15-s + (−3.95 + 0.604i)16-s + (2.80 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (0.733 + 0.679i)2-s + (0.220 + 0.350i)3-s + (0.0757 + 0.997i)4-s + (−0.173 − 0.138i)5-s + (−0.0767 + 0.406i)6-s + (−0.548 − 0.125i)7-s + (−0.622 + 0.782i)8-s + (0.359 − 0.746i)9-s + (−0.0332 − 0.219i)10-s + (0.325 − 0.929i)11-s + (−0.332 + 0.246i)12-s + (0.169 + 0.351i)13-s + (−0.316 − 0.464i)14-s + (0.0102 − 0.0913i)15-s + (−0.988 + 0.151i)16-s + (0.679 + 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22670 + 0.829745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22670 + 0.829745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.03 - 0.961i)T \) |
| 29 | \( 1 + (3.77 - 3.84i)T \) |
| good | 3 | \( 1 + (-0.381 - 0.606i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (0.388 + 0.309i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (1.45 + 0.331i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (-1.07 + 3.08i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.610 - 1.26i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-2.80 - 2.80i)T + 17iT^{2} \) |
| 19 | \( 1 + (4.35 + 2.73i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-0.0143 + 0.0114i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.262 - 2.32i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (1.69 - 0.592i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (0.378 - 0.378i)T - 41iT^{2} \) |
| 43 | \( 1 + (3.80 + 0.429i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (-8.34 - 2.92i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (6.44 - 8.07i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 - 14.5iT - 59T^{2} \) |
| 61 | \( 1 + (-0.184 + 0.116i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (-1.44 - 0.695i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 5.30i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.61 - 0.745i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (11.7 - 4.10i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (-3.90 + 0.892i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-15.4 + 1.74i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-8.35 - 5.25i)T + (42.0 + 87.3i)T^{2} \) |
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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
−13.88375019344348553471808378560, −12.82227314395935590512529712286, −12.03624946005720292097796107015, −10.70975833602916864609838204317, −9.216402068519151328783976490396, −8.353531996412876243991698128995, −6.86019321501103044179562503685, −5.96914666080147699709668184739, −4.29655131761876825369040810952, −3.31741976212892889508717250223,
2.06558539952046943745299199582, 3.67899850421372048123041399718, 5.11885682947743076363168292005, 6.54428666702854622551873760565, 7.75128860729808631477646609051, 9.458044461733806357590638990615, 10.30771341140933516075238417059, 11.46057070148825730431387996853, 12.56365294768499641476262908003, 13.12694575174062378158415995422
