| L(s) = 1 | + (0.568 + 0.822i)2-s + (−3.25 − 0.802i)3-s + (−0.354 + 0.935i)4-s + (2.16 − 1.91i)5-s + (−1.18 − 3.13i)6-s + (3.68 + 0.908i)7-s + (−0.970 + 0.239i)8-s + (7.30 + 3.83i)9-s + (2.80 + 0.692i)10-s + (−1.50 − 1.32i)11-s + (1.90 − 2.76i)12-s + (1.74 − 4.59i)13-s + (1.34 + 3.54i)14-s + (−8.59 + 4.51i)15-s + (−0.748 − 0.663i)16-s + (0.000931 − 0.00245i)17-s + ⋯ |
| L(s) = 1 | + (0.401 + 0.581i)2-s + (−1.88 − 0.463i)3-s + (−0.177 + 0.467i)4-s + (0.968 − 0.858i)5-s + (−0.485 − 1.28i)6-s + (1.39 + 0.343i)7-s + (−0.343 + 0.0846i)8-s + (2.43 + 1.27i)9-s + (0.888 + 0.218i)10-s + (−0.452 − 0.400i)11-s + (0.550 − 0.796i)12-s + (0.483 − 1.27i)13-s + (0.359 + 0.948i)14-s + (−2.21 + 1.16i)15-s + (−0.187 − 0.165i)16-s + (0.000225 − 0.000595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01497 + 0.00864538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01497 + 0.00864538i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.568 - 0.822i)T \) |
| 79 | \( 1 + (4.51 - 7.65i)T \) |
| good | 3 | \( 1 + (3.25 + 0.802i)T + (2.65 + 1.39i)T^{2} \) |
| 5 | \( 1 + (-2.16 + 1.91i)T + (0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (-3.68 - 0.908i)T + (6.19 + 3.25i)T^{2} \) |
| 11 | \( 1 + (1.50 + 1.32i)T + (1.32 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 4.59i)T + (-9.73 - 8.62i)T^{2} \) |
| 17 | \( 1 + (-0.000931 + 0.00245i)T + (-12.7 - 11.2i)T^{2} \) |
| 19 | \( 1 + (-0.346 - 2.85i)T + (-18.4 + 4.54i)T^{2} \) |
| 23 | \( 1 - 1.80T + 23T^{2} \) |
| 29 | \( 1 + (4.39 - 2.30i)T + (16.4 - 23.8i)T^{2} \) |
| 31 | \( 1 + (-0.121 - 0.176i)T + (-10.9 + 28.9i)T^{2} \) |
| 37 | \( 1 + (-0.517 - 4.26i)T + (-35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (0.227 - 0.201i)T + (4.94 - 40.7i)T^{2} \) |
| 43 | \( 1 + (1.43 - 1.27i)T + (5.18 - 42.6i)T^{2} \) |
| 47 | \( 1 + (-0.668 + 5.50i)T + (-45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (5.35 - 1.32i)T + (46.9 - 24.6i)T^{2} \) |
| 59 | \( 1 + (-0.620 - 1.63i)T + (-44.1 + 39.1i)T^{2} \) |
| 61 | \( 1 + (-1.69 - 13.9i)T + (-59.2 + 14.5i)T^{2} \) |
| 67 | \( 1 + (-0.913 + 1.32i)T + (-23.7 - 62.6i)T^{2} \) |
| 71 | \( 1 + (14.0 - 3.47i)T + (62.8 - 32.9i)T^{2} \) |
| 73 | \( 1 + (2.62 + 6.91i)T + (-54.6 + 48.4i)T^{2} \) |
| 83 | \( 1 + (-5.15 + 13.5i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (17.4 + 4.30i)T + (78.8 + 41.3i)T^{2} \) |
| 97 | \( 1 + (-1.50 - 12.3i)T + (-94.1 + 23.2i)T^{2} \) |
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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
−12.96284680631694575166081664314, −12.03804598971360305697355749422, −11.12943220749279068457363912495, −10.20218312760665332298141525987, −8.498116176665898794465129602175, −7.47229445074896449999946296668, −5.90946995093452125440176229964, −5.50439493628024032212249296326, −4.76275023288533086317758435120, −1.40251216539288017169495682873,
1.73789031808656633978286963124, 4.27809957048363002064876000186, 5.14299667748576046600567439624, 6.16750778415547117082225602382, 7.15085549635736328979129255590, 9.433721680612967114219163963481, 10.34802618323440762932581909261, 11.22844335930926360522998404576, 11.35269054090405426210340398014, 12.69023470079995901156231921148
