| L(s) = 1 | + (0.114 − 0.331i)2-s + (0.458 + 0.888i)3-s + (1.47 + 1.16i)4-s + (0.530 − 0.0506i)5-s + (0.347 − 0.0499i)6-s + (0.390 − 2.61i)7-s + (1.14 − 0.735i)8-s + (−0.580 + 0.814i)9-s + (0.0440 − 0.181i)10-s + (3.27 − 1.13i)11-s + (−0.355 + 1.84i)12-s + (−1.53 − 5.22i)13-s + (−0.823 − 0.429i)14-s + (0.288 + 0.448i)15-s + (0.772 + 3.18i)16-s + (4.87 + 1.95i)17-s + ⋯ |
| L(s) = 1 | + (0.0811 − 0.234i)2-s + (0.264 + 0.513i)3-s + (0.737 + 0.580i)4-s + (0.237 − 0.0226i)5-s + (0.141 − 0.0203i)6-s + (0.147 − 0.989i)7-s + (0.404 − 0.260i)8-s + (−0.193 + 0.271i)9-s + (0.0139 − 0.0574i)10-s + (0.988 − 0.341i)11-s + (−0.102 + 0.532i)12-s + (−0.425 − 1.44i)13-s + (−0.219 − 0.114i)14-s + (0.0743 + 0.115i)15-s + (0.193 + 0.795i)16-s + (1.18 + 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06117 + 0.106027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06117 + 0.106027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.458 - 0.888i)T \) |
| 7 | \( 1 + (-0.390 + 2.61i)T \) |
| 23 | \( 1 + (3.61 - 3.15i)T \) |
| good | 2 | \( 1 + (-0.114 + 0.331i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (-0.530 + 0.0506i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-3.27 + 1.13i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (1.53 + 5.22i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-4.87 - 1.95i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (0.705 - 0.282i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-1.32 - 9.18i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (5.85 - 0.278i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-1.19 - 0.850i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (0.252 + 0.115i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.75 - 2.73i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-3.80 + 2.19i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.74 + 8.12i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-1.78 - 0.432i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (6.19 + 3.19i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-0.663 - 3.44i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (4.66 + 5.38i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-9.94 + 12.6i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-8.92 + 9.36i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.76 + 3.85i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.703 - 14.7i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (1.63 - 3.57i)T + (-63.5 - 73.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80523464213608781021251878384, −10.33349527453379520588999048207, −9.390044500665823469134597894439, −8.046635696133681276667400772433, −7.59820382265513058231520815998, −6.40349365798334602104475427464, −5.26221344673282936718912907590, −3.75603700154831848350245740424, −3.31466272326380637095765907277, −1.57218397434710998920938486951,
1.69084169416877724376019630784, 2.48225704847251057910173553966, 4.26656246280773626663588426174, 5.65055014089918987197106379702, 6.31567561551106959023650374215, 7.17308970096671730122640053104, 8.127329551289142080304924783376, 9.380575121420076769055078853727, 9.775123051234942606948469778458, 11.22243087151863649408598693209
