| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (0.459 + 1.67i)3-s + (0.984 + 0.173i)4-s + (−2.23 − 0.107i)5-s + (−0.312 − 1.70i)6-s + (−1.02 − 0.721i)7-s + (−0.965 − 0.258i)8-s + (−2.57 + 1.53i)9-s + (2.21 + 0.302i)10-s + (−2.11 + 5.81i)11-s + (0.162 + 1.72i)12-s + (−0.354 − 4.04i)13-s + (0.963 + 0.808i)14-s + (−0.846 − 3.77i)15-s + (0.939 + 0.342i)16-s + (−1.07 + 0.287i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.265 + 0.964i)3-s + (0.492 + 0.0868i)4-s + (−0.998 − 0.0482i)5-s + (−0.127 − 0.695i)6-s + (−0.389 − 0.272i)7-s + (−0.341 − 0.0915i)8-s + (−0.859 + 0.511i)9-s + (0.700 + 0.0955i)10-s + (−0.637 + 1.75i)11-s + (0.0469 + 0.497i)12-s + (−0.0982 − 1.12i)13-s + (0.257 + 0.216i)14-s + (−0.218 − 0.975i)15-s + (0.234 + 0.0855i)16-s + (−0.260 + 0.0698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0309546 + 0.353833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0309546 + 0.353833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 + (-0.459 - 1.67i)T \) |
| 5 | \( 1 + (2.23 + 0.107i)T \) |
| good | 7 | \( 1 + (1.02 + 0.721i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (2.11 - 5.81i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.354 + 4.04i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.07 - 0.287i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.49 - 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.75 + 3.93i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.572 + 0.480i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.766 - 4.34i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.29 - 4.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (3.72 - 4.43i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.20 - 11.1i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.919 + 1.31i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (0.785 - 0.785i)T - 53iT^{2} \) |
| 59 | \( 1 + (-13.8 + 5.03i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.115 + 0.653i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.62 - 0.317i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-9.91 - 5.72i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.410 - 1.53i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.82 - 5.74i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.846 - 9.67i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (0.959 + 1.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.12 + 1.45i)T + (62.3 - 74.3i)T^{2} \) |
| 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
−12.31087751552513282780668050348, −11.08399329938733849460882360542, −10.25636610636599922207488899951, −9.797209389348181589825224686761, −8.394592707223629892650511882633, −7.897167386541578042145519721343, −6.68411827452824663370697690363, −4.98033549110376322725395695834, −3.97154597842396610796302916503, −2.62454268700022192849792986736,
0.30993204734029730618284459289, 2.40743766971324214125279472850, 3.71793943653912669207109403010, 5.77211042915921914841994785667, 6.75630486833895612073629853086, 7.66629837498445791363078667649, 8.574548609851691439494946869651, 9.101766876575202338601141679933, 10.79494816399835230079649744289, 11.44200166086505728710371059183
