| L(s) = 1 | + (−0.818 − 0.525i)2-s + (0.989 − 0.142i)3-s + (−0.437 − 0.958i)4-s + (0.929 + 0.272i)5-s + (−0.884 − 0.404i)6-s + (−0.0253 + 2.64i)7-s + (−0.422 + 2.94i)8-s + (0.959 − 0.281i)9-s + (−0.617 − 0.712i)10-s + (2.79 + 4.34i)11-s + (−0.569 − 0.886i)12-s + (−0.940 + 0.815i)13-s + (1.41 − 2.15i)14-s + (0.958 + 0.137i)15-s + (0.512 − 0.591i)16-s + (−0.140 + 0.307i)17-s + ⋯ |
| L(s) = 1 | + (−0.578 − 0.371i)2-s + (0.571 − 0.0821i)3-s + (−0.218 − 0.479i)4-s + (0.415 + 0.122i)5-s + (−0.361 − 0.164i)6-s + (−0.00959 + 0.999i)7-s + (−0.149 + 1.03i)8-s + (0.319 − 0.0939i)9-s + (−0.195 − 0.225i)10-s + (0.842 + 1.31i)11-s + (−0.164 − 0.255i)12-s + (−0.260 + 0.226i)13-s + (0.377 − 0.575i)14-s + (0.247 + 0.0355i)15-s + (0.128 − 0.147i)16-s + (−0.0341 + 0.0746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26558 + 0.123675i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26558 + 0.123675i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 7 | \( 1 + (0.0253 - 2.64i)T \) |
| 23 | \( 1 + (-4.07 + 2.52i)T \) |
| good | 2 | \( 1 + (0.818 + 0.525i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-0.929 - 0.272i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 4.34i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.940 - 0.815i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.140 - 0.307i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.62 - 5.74i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.59 + 5.68i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (6.95 + 0.999i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.359 + 1.22i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.06 + 3.62i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-2.84 + 0.409i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 8.50iT - 47T^{2} \) |
| 53 | \( 1 + (-6.02 - 5.21i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.94 + 6.01i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.508 - 3.53i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.35 + 5.21i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (5.04 + 3.24i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.57 + 0.721i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.46 + 2.13i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.97 + 1.75i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.32 - 9.22i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (14.8 + 4.35i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.79642871959696858354007569054, −9.666607675176324340778702275548, −9.544117901909710130345503201780, −8.607498633674989382323992571630, −7.59025438034588307557376523357, −6.34528116549174422068681187087, −5.40921533249012683835873004500, −4.18273070708357329225266983361, −2.46381365035374015670502352703, −1.66797347166594847990497820260,
0.984030941054028132146993393905, 3.13304771516866148124112578626, 3.89897819577959178531689700554, 5.27947265791627088637720304847, 6.79950922626795292801933914577, 7.30471474570457808603470665492, 8.403666219178127550157818414758, 9.088446569058366652968807484683, 9.704859600456280356314790561759, 10.81772902319387687433945123476
