| L(s) = 1 | + (−0.461 − 1.33i)2-s + (0.888 + 0.458i)3-s + (0.00994 − 0.00782i)4-s + (3.91 + 0.373i)5-s + (0.200 − 1.39i)6-s + (−1.00 − 2.44i)7-s + (−2.38 − 1.53i)8-s + (0.580 + 0.814i)9-s + (−1.30 − 5.38i)10-s + (0.718 − 2.07i)11-s + (0.0124 − 0.00239i)12-s + (−2.77 − 0.815i)13-s + (−2.79 + 2.46i)14-s + (3.30 + 2.12i)15-s + (−0.937 + 3.86i)16-s + (−5.91 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.326 − 0.942i)2-s + (0.513 + 0.264i)3-s + (0.00497 − 0.00391i)4-s + (1.75 + 0.167i)5-s + (0.0819 − 0.569i)6-s + (−0.380 − 0.924i)7-s + (−0.843 − 0.542i)8-s + (0.193 + 0.271i)9-s + (−0.413 − 1.70i)10-s + (0.216 − 0.625i)11-s + (0.00358 − 0.000691i)12-s + (−0.770 − 0.226i)13-s + (−0.747 + 0.659i)14-s + (0.854 + 0.549i)15-s + (−0.234 + 0.965i)16-s + (−1.43 + 0.574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0463 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0463 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27074 - 1.33103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27074 - 1.33103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.888 - 0.458i)T \) |
| 7 | \( 1 + (1.00 + 2.44i)T \) |
| 23 | \( 1 + (-3.80 + 2.91i)T \) |
| good | 2 | \( 1 + (0.461 + 1.33i)T + (-1.57 + 1.23i)T^{2} \) |
| 5 | \( 1 + (-3.91 - 0.373i)T + (4.90 + 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.718 + 2.07i)T + (-8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (2.77 + 0.815i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (5.91 - 2.36i)T + (12.3 - 11.7i)T^{2} \) |
| 19 | \( 1 + (-5.31 - 2.12i)T + (13.7 + 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.814 + 5.66i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.517 - 10.8i)T + (-30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-0.577 - 0.810i)T + (-12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-3.70 - 8.12i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.57 - 1.65i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (1.20 - 2.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.89 + 7.53i)T + (2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (1.16 + 4.79i)T + (-52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-1.87 + 0.964i)T + (35.3 - 49.6i)T^{2} \) |
| 67 | \( 1 + (-13.2 - 2.55i)T + (62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-7.11 + 8.21i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (2.17 - 1.70i)T + (17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (3.91 - 3.72i)T + (3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (2.79 - 6.12i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.0143 + 0.300i)T + (-88.5 + 8.45i)T^{2} \) |
| 97 | \( 1 + (0.168 + 0.369i)T + (-63.5 + 73.3i)T^{2} \) |
| 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
−10.62416356712897959713844160197, −9.738861052673273821358632491276, −9.579296595811400614722487278548, −8.438690275268987149336280132330, −6.84424065326723047771124310468, −6.25825683523702896426472873575, −4.93769462871572009909369269597, −3.36224357010062323301241877513, −2.49692390662229280139390418581, −1.30561395905565905534975976155,
2.10792948619898622667123551108, 2.76686132972809965194560845412, 5.02485041491486888926414155328, 5.79293626861169418241014183936, 6.76568997173886839675472006911, 7.30467718813243690521865915166, 8.728819380518013985843467069648, 9.412157150796073127943745495430, 9.572076788094491754025490357448, 11.23921909334219090144068615838
