| L(s) = 1 | + (0.327 + 0.460i)2-s + (−0.235 − 0.971i)3-s + (0.549 − 1.58i)4-s + (−0.198 − 4.17i)5-s + (0.369 − 0.426i)6-s + (2.54 − 0.738i)7-s + (1.99 − 0.585i)8-s + (−0.888 + 0.458i)9-s + (1.85 − 1.45i)10-s + (−3.13 + 4.40i)11-s + (−1.67 − 0.159i)12-s + (0.0787 + 0.547i)13-s + (1.17 + 0.926i)14-s + (−4.00 + 1.17i)15-s + (−1.71 − 1.35i)16-s + (−1.13 − 0.219i)17-s + ⋯ |
| L(s) = 1 | + (0.231 + 0.325i)2-s + (−0.136 − 0.561i)3-s + (0.274 − 0.794i)4-s + (−0.0889 − 1.86i)5-s + (0.150 − 0.174i)6-s + (0.960 − 0.279i)7-s + (0.705 − 0.207i)8-s + (−0.296 + 0.152i)9-s + (0.586 − 0.461i)10-s + (−0.946 + 1.32i)11-s + (−0.483 − 0.0461i)12-s + (0.0218 + 0.151i)13-s + (0.313 + 0.247i)14-s + (−1.03 + 0.303i)15-s + (−0.429 − 0.338i)16-s + (−0.276 − 0.0532i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02197 - 1.32649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02197 - 1.32649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.235 + 0.971i)T \) |
| 7 | \( 1 + (-2.54 + 0.738i)T \) |
| 23 | \( 1 + (-3.87 - 2.82i)T \) |
| good | 2 | \( 1 + (-0.327 - 0.460i)T + (-0.654 + 1.89i)T^{2} \) |
| 5 | \( 1 + (0.198 + 4.17i)T + (-4.97 + 0.475i)T^{2} \) |
| 11 | \( 1 + (3.13 - 4.40i)T + (-3.59 - 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.0787 - 0.547i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.13 + 0.219i)T + (15.7 + 6.31i)T^{2} \) |
| 19 | \( 1 + (-3.13 + 0.605i)T + (17.6 - 7.06i)T^{2} \) |
| 29 | \( 1 + (2.55 - 2.94i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-3.98 - 3.80i)T + (1.47 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-8.01 + 4.12i)T + (21.4 - 30.1i)T^{2} \) |
| 41 | \( 1 + (-2.12 + 1.36i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.539 - 0.158i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-3.26 + 5.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (12.0 - 4.84i)T + (38.3 - 36.5i)T^{2} \) |
| 59 | \( 1 + (0.280 - 0.220i)T + (13.9 - 57.3i)T^{2} \) |
| 61 | \( 1 + (2.36 - 9.74i)T + (-54.2 - 27.9i)T^{2} \) |
| 67 | \( 1 + (-5.66 + 0.541i)T + (65.7 - 12.6i)T^{2} \) |
| 71 | \( 1 + (-4.93 + 10.8i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.79 + 8.08i)T + (-57.3 - 45.1i)T^{2} \) |
| 79 | \( 1 + (7.82 + 3.13i)T + (57.1 + 54.5i)T^{2} \) |
| 83 | \( 1 + (12.0 + 7.73i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.39 + 1.33i)T + (4.23 - 88.8i)T^{2} \) |
| 97 | \( 1 + (-8.34 + 5.36i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.82989069198720012929470387233, −9.722738750995847825987287654052, −8.900797157661527714387065067334, −7.74108185388470498432919033663, −7.28565196020678133000481860214, −5.75986274204645589888901856288, −4.94256147085616132159921493457, −4.58427862316011856797783271939, −1.96672529800461685876495755951, −1.03813014014035838001265481653,
2.58832243670118253284841885800, 3.08178432335519814497266578201, 4.26134452091224374007728318601, 5.60808175075464898039902321622, 6.63319491103454572275548524671, 7.81100772760447201256950882536, 8.204928371240376995578638115826, 9.745850758874196455102959556751, 10.85754179110140695132247579387, 11.14518069853191940610430343024
