| L(s) = 1 | − 0.670·2-s + 3-s − 1.54·4-s + 2.54·5-s − 0.670·6-s + 7-s + 2.38·8-s + 9-s − 1.71·10-s − 3.05·11-s − 1.54·12-s + 13-s − 0.670·14-s + 2.54·15-s + 1.50·16-s + 1.34·17-s − 0.670·18-s + 7.60·19-s − 3.95·20-s + 21-s + 2.04·22-s + 1.84·23-s + 2.38·24-s + 1.50·25-s − 0.670·26-s + 27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | − 0.474·2-s + 0.577·3-s − 0.774·4-s + 1.14·5-s − 0.273·6-s + 0.377·7-s + 0.841·8-s + 0.333·9-s − 0.540·10-s − 0.920·11-s − 0.447·12-s + 0.277·13-s − 0.179·14-s + 0.658·15-s + 0.375·16-s + 0.325·17-s − 0.158·18-s + 1.74·19-s − 0.883·20-s + 0.218·21-s + 0.436·22-s + 0.384·23-s + 0.486·24-s + 0.300·25-s − 0.131·26-s + 0.192·27-s − 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.266716748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.266716748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.670T + 2T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 17 | \( 1 - 1.34T + 17T^{2} \) |
| 19 | \( 1 - 7.60T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 9.20T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 - 0.502T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 + 0.683T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.91T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
| 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
−11.93231658786834200383615635081, −10.48008047307770936312544198274, −9.908057874970054667782797656053, −9.113303479787044573768373045809, −8.207223953076670859631588495581, −7.31828630396094412973955876395, −5.66972635446158703455868932222, −4.83973347330053675956896965071, −3.17005981994733725362425823749, −1.53473120245872328500297646797,
1.53473120245872328500297646797, 3.17005981994733725362425823749, 4.83973347330053675956896965071, 5.66972635446158703455868932222, 7.31828630396094412973955876395, 8.207223953076670859631588495581, 9.113303479787044573768373045809, 9.908057874970054667782797656053, 10.48008047307770936312544198274, 11.93231658786834200383615635081
