| L(s) = 1 | + (0.669 + 0.743i)2-s + (0.825 + 1.52i)3-s + (−0.104 + 0.994i)4-s + (−1.65 − 1.48i)5-s + (−0.578 + 1.63i)6-s + (1.13 + 2.55i)7-s + (−0.809 + 0.587i)8-s + (−1.63 + 2.51i)9-s − 2.22i·10-s + (3.29 − 0.366i)11-s + (−1.60 + 0.662i)12-s + (0.415 + 1.95i)13-s + (−1.13 + 2.55i)14-s + (0.899 − 3.73i)15-s + (−0.978 − 0.207i)16-s + (−1.64 − 5.07i)17-s + ⋯ |
| L(s) = 1 | + (0.473 + 0.525i)2-s + (0.476 + 0.879i)3-s + (−0.0522 + 0.497i)4-s + (−0.738 − 0.664i)5-s + (−0.236 + 0.666i)6-s + (0.429 + 0.965i)7-s + (−0.286 + 0.207i)8-s + (−0.545 + 0.838i)9-s − 0.702i·10-s + (0.993 − 0.110i)11-s + (−0.462 + 0.191i)12-s + (0.115 + 0.542i)13-s + (−0.303 + 0.682i)14-s + (0.232 − 0.965i)15-s + (−0.244 − 0.0519i)16-s + (−0.399 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0705 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0705 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08180 + 1.16097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08180 + 1.16097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (-0.825 - 1.52i)T \) |
| 11 | \( 1 + (-3.29 + 0.366i)T \) |
| good | 5 | \( 1 + (1.65 + 1.48i)T + (0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (-1.13 - 2.55i)T + (-4.68 + 5.20i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 1.95i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (1.64 + 5.07i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.47 + 3.40i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-6.83 + 3.94i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.26 + 0.561i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (0.773 - 0.164i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-2.71 - 1.97i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.80 - 2.14i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-7.35 - 4.24i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.37 - 0.670i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (12.3 + 3.99i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.74 + 1.02i)T + (57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (2.69 - 12.6i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (5.11 + 8.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.98 - 2.26i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.09 + 8.38i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.63 - 2.37i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-6.00 - 1.27i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 - 5.44iT - 89T^{2} \) |
| 97 | \( 1 + (1.16 + 1.28i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−12.76225017595247259711713954562, −11.71846637744317845914189937609, −11.11156529243407368452304776924, −9.095216736958293172237747815987, −9.012992893362923474956181220520, −7.85270302719093149370866103602, −6.42736990883842521299203250299, −4.87070642203658766424152240113, −4.41193356466072455573288098750, −2.81705745839703961006457545918,
1.46076249932571820947629127411, 3.29645749747255290237044978873, 4.15368902161662680893166522818, 6.11006108317979580222877703759, 7.14494778631970287390431395244, 7.974549866044688409416954896871, 9.219323325771086346806066134620, 10.73028692710831454598359379628, 11.22323414688105094154722531889, 12.36813785699057509025449391014
