| L(s) = 1 | + (−0.428 + 0.903i)2-s + (−0.575 − 1.98i)3-s + (−0.632 − 0.774i)4-s + (0.764 + 3.74i)5-s + (2.04 + 0.331i)6-s + (1.67 − 1.74i)7-s + (0.970 − 0.239i)8-s + (−1.07 + 0.682i)9-s + (−3.71 − 0.915i)10-s + (4.09 − 1.36i)11-s + (−1.17 + 1.70i)12-s + (3.80 − 0.619i)13-s + (0.856 + 2.25i)14-s + (7.00 − 3.67i)15-s + (−0.200 + 0.979i)16-s + (−0.870 + 2.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.303 + 0.638i)2-s + (−0.332 − 1.14i)3-s + (−0.316 − 0.387i)4-s + (0.342 + 1.67i)5-s + (0.833 + 0.135i)6-s + (0.632 − 0.658i)7-s + (0.343 − 0.0846i)8-s + (−0.359 + 0.227i)9-s + (−1.17 − 0.289i)10-s + (1.23 − 0.412i)11-s + (−0.339 + 0.491i)12-s + (1.05 − 0.171i)13-s + (0.228 + 0.603i)14-s + (1.80 − 0.948i)15-s + (−0.0500 + 0.244i)16-s + (−0.211 + 0.556i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.999582 + 0.112226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999582 + 0.112226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.428 - 0.903i)T \) |
| 79 | \( 1 + (-8.18 - 3.46i)T \) |
| good | 3 | \( 1 + (0.575 + 1.98i)T + (-2.53 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.764 - 3.74i)T + (-4.59 + 1.95i)T^{2} \) |
| 7 | \( 1 + (-1.67 + 1.74i)T + (-0.281 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-4.09 + 1.36i)T + (8.79 - 6.60i)T^{2} \) |
| 13 | \( 1 + (-3.80 + 0.619i)T + (12.3 - 4.11i)T^{2} \) |
| 17 | \( 1 + (0.870 - 2.29i)T + (-12.7 - 11.2i)T^{2} \) |
| 19 | \( 1 + (5.09 + 2.16i)T + (13.1 + 13.7i)T^{2} \) |
| 23 | \( 1 + (3.64 - 6.32i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0655 + 1.62i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-6.76 + 0.546i)T + (30.5 - 4.97i)T^{2} \) |
| 37 | \( 1 + (5.46 - 4.10i)T + (10.2 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-3.56 + 3.15i)T + (4.94 - 40.7i)T^{2} \) |
| 43 | \( 1 + (3.14 + 1.05i)T + (34.3 + 25.8i)T^{2} \) |
| 47 | \( 1 + (5.22 + 3.92i)T + (13.0 + 45.1i)T^{2} \) |
| 53 | \( 1 + (-2.39 + 8.25i)T + (-44.7 - 28.3i)T^{2} \) |
| 59 | \( 1 + (4.38 - 5.37i)T + (-11.8 - 57.8i)T^{2} \) |
| 61 | \( 1 + (1.41 + 11.6i)T + (-59.2 + 14.5i)T^{2} \) |
| 67 | \( 1 + (6.21 - 8.99i)T + (-23.7 - 62.6i)T^{2} \) |
| 71 | \( 1 + (1.66 - 0.409i)T + (62.8 - 32.9i)T^{2} \) |
| 73 | \( 1 + (7.71 + 1.25i)T + (69.2 + 23.1i)T^{2} \) |
| 83 | \( 1 + (4.24 + 5.19i)T + (-16.6 + 81.3i)T^{2} \) |
| 89 | \( 1 + (0.557 + 0.137i)T + (78.8 + 41.3i)T^{2} \) |
| 97 | \( 1 + (1.01 + 8.32i)T + (-94.1 + 23.2i)T^{2} \) |
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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
−13.40739345108896113994388553455, −11.71939550061113103151245488943, −11.01121324981574856363760507969, −10.05701108174536435905316735074, −8.470811062221604155640073614753, −7.42297931003128294319975495432, −6.51023698509504972191024330084, −6.15424305959445492307277917294, −3.82900118904437600697729809934, −1.63857903604333517125033397016,
1.63941658890129057860198841598, 4.20245394286564345645793552149, 4.70214048540529175055775857173, 6.05398259435642419158453359921, 8.426423032909745072325714084899, 8.899346804806432102437301470928, 9.755120436312674806674086722484, 10.82407665078333646297121800215, 11.90272541617636532244373332478, 12.49805326148318002090675663893
