| L(s) = 1 | + (−1.41 − 0.0616i)2-s + (−0.0537 − 0.0699i)3-s + (1.99 + 0.174i)4-s + (−0.644 − 0.494i)5-s + (0.0715 + 0.102i)6-s + (−2.22 − 1.43i)7-s + (−2.80 − 0.368i)8-s + (0.774 − 2.89i)9-s + (0.880 + 0.739i)10-s + (−1.22 − 0.160i)11-s + (−0.0948 − 0.148i)12-s + (3.26 + 1.35i)13-s + (3.05 + 2.15i)14-s + 0.0717i·15-s + (3.93 + 0.694i)16-s + (−4.78 − 2.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0435i)2-s + (−0.0310 − 0.0404i)3-s + (0.996 + 0.0871i)4-s + (−0.288 − 0.221i)5-s + (0.0292 + 0.0417i)6-s + (−0.841 − 0.540i)7-s + (−0.991 − 0.130i)8-s + (0.258 − 0.963i)9-s + (0.278 + 0.233i)10-s + (−0.368 − 0.0484i)11-s + (−0.0273 − 0.0429i)12-s + (0.905 + 0.374i)13-s + (0.816 + 0.577i)14-s + 0.0185i·15-s + (0.984 + 0.173i)16-s + (−1.15 − 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.334813 - 0.437305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.334813 - 0.437305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0616i)T \) |
| 7 | \( 1 + (2.22 + 1.43i)T \) |
| good | 3 | \( 1 + (0.0537 + 0.0699i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (0.644 + 0.494i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.160i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (-3.26 - 1.35i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (4.78 + 2.75i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.966 + 7.34i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (0.256 - 0.958i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.91 + 7.04i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (2.06 - 3.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.99 - 6.13i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-4.78 - 4.78i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.61 + 3.89i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (10.4 - 6.05i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.988 - 0.130i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.0489 + 0.372i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-2.69 + 0.354i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-7.87 - 10.2i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-0.873 + 0.873i)T - 71iT^{2} \) |
| 73 | \( 1 + (-4.29 + 1.14i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.74 + 4.47i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.84 - 1.18i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-3.45 - 0.926i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−11.66779830543404398892893183029, −11.01642689440658178169803065920, −9.795578735702728874150656736933, −9.168355588635973936659514450040, −8.149303173021154559792797190725, −6.81468818468891356904073712708, −6.38168065197739245531228608716, −4.27939923995338882028831020729, −2.82596722334851861452535274704, −0.61429233051647294836710183682,
2.08024162610194070196409127714, 3.60436363428472747837940242729, 5.62519526370977536387890467234, 6.56711813639230067322110324424, 7.79118712498014957141099668270, 8.532771628791218630792152354459, 9.639789580989743556339689253904, 10.62016449028690677240046659566, 11.15852812284624599716842707857, 12.51593369186288822229115704242
