L(s) = 1 | + 2-s + 1.13·3-s + 4-s − 5-s + 1.13·6-s − 0.848·7-s + 8-s − 1.71·9-s − 10-s + 11-s + 1.13·12-s + 0.342·13-s − 0.848·14-s − 1.13·15-s + 16-s + 0.0704·17-s − 1.71·18-s + 5.57·19-s − 20-s − 0.961·21-s + 22-s + 2.81·23-s + 1.13·24-s + 25-s + 0.342·26-s − 5.34·27-s − 0.848·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.654·3-s + 0.5·4-s − 0.447·5-s + 0.462·6-s − 0.320·7-s + 0.353·8-s − 0.571·9-s − 0.316·10-s + 0.301·11-s + 0.327·12-s + 0.0949·13-s − 0.226·14-s − 0.292·15-s + 0.250·16-s + 0.0170·17-s − 0.404·18-s + 1.27·19-s − 0.223·20-s − 0.209·21-s + 0.213·22-s + 0.587·23-s + 0.231·24-s + 0.200·25-s + 0.0671·26-s − 1.02·27-s − 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.518795288\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.518795288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 7 | \( 1 + 0.848T + 7T^{2} \) |
| 13 | \( 1 - 0.342T + 13T^{2} \) |
| 17 | \( 1 - 0.0704T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 - 0.000180T + 41T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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
−8.171059933699853186862069566786, −7.65486634323679248641422016791, −6.72803599331108823093620915553, −6.19798400839284525449053709296, −5.18828843765833972370131630382, −4.60548283458380484670800018627, −3.42626714132522268973222701484, −3.22754339639412851951838489303, −2.24919344368136615959035834800, −0.914969510696414713621043052458,
0.914969510696414713621043052458, 2.24919344368136615959035834800, 3.22754339639412851951838489303, 3.42626714132522268973222701484, 4.60548283458380484670800018627, 5.18828843765833972370131630382, 6.19798400839284525449053709296, 6.72803599331108823093620915553, 7.65486634323679248641422016791, 8.171059933699853186862069566786