| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (1.10 − 1.27i)5-s + (−0.281 + 0.959i)6-s + (1.32 + 2.28i)7-s + (0.415 + 0.909i)8-s + (0.654 + 0.755i)9-s + (−1.41 − 0.910i)10-s + (−3.28 − 0.472i)11-s + (0.989 + 0.142i)12-s + (0.697 − 1.08i)13-s + (2.07 − 1.64i)14-s + (−1.53 + 0.699i)15-s + (0.841 − 0.540i)16-s + (0.711 + 0.208i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (0.493 − 0.568i)5-s + (−0.115 + 0.391i)6-s + (0.502 + 0.864i)7-s + (0.146 + 0.321i)8-s + (0.218 + 0.251i)9-s + (−0.447 − 0.287i)10-s + (−0.990 − 0.142i)11-s + (0.285 + 0.0410i)12-s + (0.193 − 0.300i)13-s + (0.554 − 0.438i)14-s + (−0.395 + 0.180i)15-s + (0.210 − 0.135i)16-s + (0.172 + 0.0506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0826 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0826 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991896 - 0.913061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991896 - 0.913061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (-1.32 - 2.28i)T \) |
| 23 | \( 1 + (-4.41 + 1.86i)T \) |
| good | 5 | \( 1 + (-1.10 + 1.27i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (3.28 + 0.472i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.697 + 1.08i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.711 - 0.208i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-6.49 + 1.90i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (5.76 + 1.69i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-8.46 + 3.86i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 1.13i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.0528 + 0.0457i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.79 - 2.19i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 0.942iT - 47T^{2} \) |
| 53 | \( 1 + (2.45 + 3.82i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.80 + 7.48i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.39 + 11.8i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-4.18 + 0.602i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.157 - 1.09i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.11 + 7.20i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.84 + 9.09i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (5.21 + 6.01i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.54 - 5.58i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (4.56 - 5.26i)T + (-13.8 - 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.758422136864715012782661134929, −9.211100860666172426642033805658, −8.213560857647510877774963170599, −7.53851878796487061179368209810, −6.09339553370340230995431960786, −5.27044536240508933277810904461, −4.83779973548355006311391543210, −3.16622799784704709070888488423, −2.13404771203811326957508150373, −0.869622356918880035033532741760,
1.16975525805087199095499858102, 2.96342718604491587637809674799, 4.24254014945413635363760413703, 5.18503473959678881331833382734, 5.85141431941949590785812143387, 6.98364223980234812457386828978, 7.43636212301376329800499400293, 8.397672382845098147382637846409, 9.570141150791202571261047893067, 10.18204490115695957813025612147
