| L(s) = 1 | + (2.16 + 0.102i)2-s + (−0.197 + 1.72i)3-s + (2.66 + 0.254i)4-s + (0.848 − 0.809i)5-s + (−0.602 + 3.69i)6-s + (0.0842 − 0.437i)7-s + (1.45 + 0.209i)8-s + (−2.92 − 0.678i)9-s + (1.91 − 1.66i)10-s + (−1.13 − 0.583i)11-s + (−0.963 + 4.53i)12-s + (−1.03 + 0.198i)13-s + (0.227 − 0.935i)14-s + (1.22 + 1.62i)15-s + (−2.14 − 0.412i)16-s + (−0.577 + 1.26i)17-s + ⋯ |
| L(s) = 1 | + (1.52 + 0.0727i)2-s + (−0.113 + 0.993i)3-s + (1.33 + 0.127i)4-s + (0.379 − 0.361i)5-s + (−0.246 + 1.50i)6-s + (0.0318 − 0.165i)7-s + (0.513 + 0.0738i)8-s + (−0.974 − 0.226i)9-s + (0.606 − 0.525i)10-s + (−0.341 − 0.176i)11-s + (−0.278 + 1.31i)12-s + (−0.285 + 0.0550i)13-s + (0.0606 − 0.250i)14-s + (0.316 + 0.418i)15-s + (−0.535 − 0.103i)16-s + (−0.139 + 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28481 + 0.796387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28481 + 0.796387i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.197 - 1.72i)T \) |
| 23 | \( 1 + (-0.998 + 4.69i)T \) |
| good | 2 | \( 1 + (-2.16 - 0.102i)T + (1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (-0.848 + 0.809i)T + (0.237 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.0842 + 0.437i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (1.13 + 0.583i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (1.03 - 0.198i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (0.577 - 1.26i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-4.62 + 2.11i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.365 - 3.83i)T + (-28.4 + 5.48i)T^{2} \) |
| 31 | \( 1 + (4.02 + 3.16i)T + (7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.33 - 4.53i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (5.08 + 5.33i)T + (-1.95 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-0.532 - 0.677i)T + (-10.1 + 41.7i)T^{2} \) |
| 47 | \( 1 + (-0.449 + 0.259i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.31 - 1.51i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.22 + 6.38i)T + (-54.7 + 21.9i)T^{2} \) |
| 61 | \( 1 + (-2.97 - 7.43i)T + (-44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (-5.40 - 10.4i)T + (-38.8 + 54.5i)T^{2} \) |
| 71 | \( 1 + (6.34 - 9.87i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.30 - 9.41i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-13.4 - 4.64i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (5.98 + 5.71i)T + (3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (1.75 + 12.2i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (5.86 - 1.42i)T + (86.2 - 44.4i)T^{2} \) |
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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
−12.67738967875419386417967630966, −11.66532117030099699314961749268, −10.82646270620720895832478152650, −9.681685024019818415254346192135, −8.683428485831900054665380116621, −7.00343705600889280278591795033, −5.67241492347455452712308889211, −5.06642301107106837272127346065, −4.00807996277430334447215974448, −2.81705711401151934833932569723,
2.17072487229543263351542348028, 3.33032547187784215766214417981, 5.05554047161507687644482115391, 5.86442898436036913228807184990, 6.84931227097772798998448706562, 7.84362711773281988655091665172, 9.393845085884055422491330855736, 10.83005558247402365130828135614, 11.83211250188728135118977424714, 12.37220231742185335839758860493
