L(s) = 1 | − 0.923·2-s − 3-s − 1.14·4-s − 5-s + 0.923·6-s + 7-s + 2.90·8-s + 9-s + 0.923·10-s + 0.823·11-s + 1.14·12-s + 3.72·13-s − 0.923·14-s + 15-s − 0.393·16-s + 1.44·17-s − 0.923·18-s + 0.648·19-s + 1.14·20-s − 21-s − 0.761·22-s − 23-s − 2.90·24-s + 25-s − 3.44·26-s − 27-s − 1.14·28-s + ⋯ |
L(s) = 1 | − 0.653·2-s − 0.577·3-s − 0.573·4-s − 0.447·5-s + 0.377·6-s + 0.377·7-s + 1.02·8-s + 0.333·9-s + 0.292·10-s + 0.248·11-s + 0.330·12-s + 1.03·13-s − 0.246·14-s + 0.258·15-s − 0.0983·16-s + 0.350·17-s − 0.217·18-s + 0.148·19-s + 0.256·20-s − 0.218·21-s − 0.162·22-s − 0.208·23-s − 0.593·24-s + 0.200·25-s − 0.675·26-s − 0.192·27-s − 0.216·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8341403561\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8341403561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 0.923T + 2T^{2} \) |
| 11 | \( 1 - 0.823T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 + 0.382T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 + 8.19T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 - 9.64T + 97T^{2} \) |
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
−8.787805383464970984215568354865, −8.362188319805700817291004122412, −7.55869456690154058712829595676, −6.78081010889919924410100786776, −5.79986481444369732627704116194, −4.97677686425368307982821064615, −4.20434201442919585257294627343, −3.38917975884962752974233196060, −1.67309778722328275382559028918, −0.71221873729383407099206595559,
0.71221873729383407099206595559, 1.67309778722328275382559028918, 3.38917975884962752974233196060, 4.20434201442919585257294627343, 4.97677686425368307982821064615, 5.79986481444369732627704116194, 6.78081010889919924410100786776, 7.55869456690154058712829595676, 8.362188319805700817291004122412, 8.787805383464970984215568354865