L(s) = 1 | + 2-s + 3-s + 4-s + 2.22·5-s + 6-s − 0.652·7-s + 8-s + 9-s + 2.22·10-s − 1.53·11-s + 12-s + 0.467·13-s − 0.652·14-s + 2.22·15-s + 16-s − 2.10·17-s + 18-s + 2.22·20-s − 0.652·21-s − 1.53·22-s + 5.90·23-s + 24-s − 0.0418·25-s + 0.467·26-s + 27-s − 0.652·28-s + 8.33·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.995·5-s + 0.408·6-s − 0.246·7-s + 0.353·8-s + 0.333·9-s + 0.704·10-s − 0.461·11-s + 0.288·12-s + 0.129·13-s − 0.174·14-s + 0.574·15-s + 0.250·16-s − 0.510·17-s + 0.235·18-s + 0.497·20-s − 0.142·21-s − 0.326·22-s + 1.23·23-s + 0.204·24-s − 0.00837·25-s + 0.0917·26-s + 0.192·27-s − 0.123·28-s + 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.105103262\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.105103262\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 2.22T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 0.467T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 - 8.33T + 29T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 0.248T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 2.46T + 97T^{2} \) |
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
−9.081940700421271539258346796728, −8.333524126661174950034714249865, −7.41933966318204320025041704707, −6.52999800085345835332225270461, −5.99628739186168474857195042971, −4.96909058163643420547256287185, −4.30934508866633252469056170567, −2.98605416111299620352314484815, −2.55502466899488522267943121322, −1.30546881369965573227079593671,
1.30546881369965573227079593671, 2.55502466899488522267943121322, 2.98605416111299620352314484815, 4.30934508866633252469056170567, 4.96909058163643420547256287185, 5.99628739186168474857195042971, 6.52999800085345835332225270461, 7.41933966318204320025041704707, 8.333524126661174950034714249865, 9.081940700421271539258346796728