| L(s) = 1 | + (−1.74 − 0.774i)2-s + (−0.239 + 2.28i)3-s + (1.08 + 1.21i)4-s + 3.51·5-s + (2.18 − 3.78i)6-s + (1.97 + 2.18i)7-s + (0.218 + 0.672i)8-s + (−2.22 − 0.471i)9-s + (−6.10 − 2.72i)10-s + (3.45 − 0.734i)11-s + (−3.02 + 2.19i)12-s + (2.27 − 2.79i)13-s + (−1.73 − 5.33i)14-s + (−0.842 + 8.01i)15-s + (0.481 − 4.57i)16-s + (−6.08 − 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 − 0.547i)2-s + (−0.138 + 1.31i)3-s + (0.544 + 0.605i)4-s + 1.57·5-s + (0.892 − 1.54i)6-s + (0.745 + 0.827i)7-s + (0.0773 + 0.237i)8-s + (−0.740 − 0.157i)9-s + (−1.93 − 0.860i)10-s + (1.04 − 0.221i)11-s + (−0.873 + 0.634i)12-s + (0.630 − 0.776i)13-s + (−0.463 − 1.42i)14-s + (−0.217 + 2.06i)15-s + (0.120 − 1.14i)16-s + (−1.47 − 0.313i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.961453 + 0.368842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961453 + 0.368842i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.27 + 2.79i)T \) |
| 31 | \( 1 + (-2.87 + 4.76i)T \) |
| good | 2 | \( 1 + (1.74 + 0.774i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.239 - 2.28i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 + (-1.97 - 2.18i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.45 + 0.734i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (6.08 + 1.29i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.154 + 1.46i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.858 + 0.953i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.45 - 1.53i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 3.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (10.7 + 4.80i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.32 - 12.6i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (1.14 + 0.834i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.28 + 10.1i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.605 + 0.269i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (2.41 - 4.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.86 + 4.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.30 + 0.915i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (4.12 - 12.6i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.369 + 1.13i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.70 - 4.87i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (13.5 - 2.88i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (5.08 + 5.64i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−11.11655717454253704962056546970, −10.19154325271800985105906055553, −9.656025978766216594680053882187, −8.881428673719616780650433988030, −8.472624677785934643555162395304, −6.53158062352633054946380562504, −5.46875097917558922020418206291, −4.62867745862936749361294802788, −2.78047263874843241585557066628, −1.57964811317924864774221751042,
1.31642513033133429365544952505, 1.81714363569009279019354653496, 4.35685336664408197281260246383, 6.14698905902565279977906095281, 6.64320886395237615204485139603, 7.27936909324158396715258111352, 8.490330647101241614331302903953, 9.059908050351584997751461361401, 10.08433454133347253323438327104, 10.89182092694377978180245994522
