| 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 + (2.61 − 0.401i)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 + (−2.35 − 1.04i)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 + (0.988 − 0.151i)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 + (−0.628 − 0.279i)14-s + (0.247 + 0.0355i)15-s + (0.128 − 0.147i)16-s + (0.0341 − 0.0746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.684146 - 0.475102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.684146 - 0.475102i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.61 + 0.401i)T \) |
| 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
−10.74861698162722277077907379096, −10.07195220622013002066440570133, −9.104591861289330490154390691571, −8.313029554679846521892216979680, −7.21571070543992050198549748665, −6.16116530100204524041537676011, −4.77868346473336543173164136433, −4.43257112560158570585836010336, −2.20128320365436962947568010244, −0.848565432666313795377492637830,
1.20381825448030663803722794962, 3.42925154721121354682085809862, 4.35831665150037436549552293739, 5.72195563141046351734697325011, 6.65422005831920567483260913940, 7.69560680052912959123076118207, 8.404885191560888713686008039443, 9.064719394882558327223187686897, 10.31273449755577047820084398540, 11.29119429058391430517247019623
