| L(s) = 1 | + (−1.04 − 2.05i)2-s + (1.12 − 1.32i)3-s + (−1.95 + 2.68i)4-s + (−0.262 − 1.65i)5-s + (−3.88 − 0.920i)6-s + (−0.575 + 0.491i)7-s + (3.01 + 0.477i)8-s + (−0.488 − 2.95i)9-s + (−3.13 + 2.27i)10-s + (2.56 + 1.57i)11-s + (1.36 + 5.59i)12-s + (0.892 − 0.0702i)13-s + (1.61 + 0.668i)14-s + (−2.48 − 1.51i)15-s + (−0.123 − 0.381i)16-s + (−6.64 + 1.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.740 − 1.45i)2-s + (0.646 − 0.762i)3-s + (−0.976 + 1.34i)4-s + (−0.117 − 0.741i)5-s + (−1.58 − 0.375i)6-s + (−0.217 + 0.185i)7-s + (1.06 + 0.168i)8-s + (−0.162 − 0.986i)9-s + (−0.991 + 0.720i)10-s + (0.772 + 0.473i)11-s + (0.393 + 1.61i)12-s + (0.247 − 0.0194i)13-s + (0.431 + 0.178i)14-s + (−0.641 − 0.390i)15-s + (−0.0309 − 0.0953i)16-s + (−1.61 + 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112738 - 0.822148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112738 - 0.822148i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.12 + 1.32i)T \) |
| 41 | \( 1 + (-2.26 - 5.99i)T \) |
| good | 2 | \( 1 + (1.04 + 2.05i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.262 + 1.65i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (0.575 - 0.491i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 1.57i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.892 + 0.0702i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (6.64 - 1.59i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-6.20 - 0.488i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-1.15 + 3.54i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-8.22 - 1.97i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (5.18 + 7.13i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.171 + 0.124i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-1.33 + 0.679i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (5.37 - 6.29i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (0.608 - 2.53i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (0.108 + 0.0350i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.195 + 0.383i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (6.76 - 4.14i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (0.826 - 1.34i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-8.35 - 8.35i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.126 - 0.0525i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 2.84iT - 83T^{2} \) |
| 89 | \( 1 + (-4.76 - 5.57i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (-9.09 - 14.8i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−12.67282782746936972961846015753, −12.02515839645545716436301277318, −11.02968821443947850895350314388, −9.529170886275332400053195753594, −9.002053052329715364276169798985, −8.077670381214922428404873870170, −6.54103295622063891530590921963, −4.22746580220979204017801443711, −2.71537073678242299236143680157, −1.23350298128916107738051545017,
3.32617455158350068041151334803, 5.04985683219801243889747915816, 6.55789222841753351548686071236, 7.36786745889639559739228655694, 8.670677909757818788036940601628, 9.270105273024243291705521449085, 10.38668122445558419679736379444, 11.47925247977332784458988800248, 13.63181060924883608757074643605, 14.18458343188355183755796811196
