| 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
−11.29119429058391430517247019623, −10.31273449755577047820084398540, −9.064719394882558327223187686897, −8.404885191560888713686008039443, −7.69560680052912959123076118207, −6.65422005831920567483260913940, −5.72195563141046351734697325011, −4.35831665150037436549552293739, −3.42925154721121354682085809862, −1.20381825448030663803722794962,
0.848565432666313795377492637830, 2.20128320365436962947568010244, 4.43257112560158570585836010336, 4.77868346473336543173164136433, 6.16116530100204524041537676011, 7.21571070543992050198549748665, 8.313029554679846521892216979680, 9.104591861289330490154390691571, 10.07195220622013002066440570133, 10.74861698162722277077907379096
