L(s) = 1 | − 1.71·2-s + 3.15·3-s + 0.938·4-s − 2.70·5-s − 5.40·6-s + 1.61·7-s + 1.81·8-s + 6.93·9-s + 4.64·10-s + 0.692·11-s + 2.95·12-s + 1.52·13-s − 2.76·14-s − 8.53·15-s − 4.99·16-s + 1.05·17-s − 11.8·18-s + 0.128·19-s − 2.54·20-s + 5.08·21-s − 1.18·22-s + 1.88·23-s + 5.73·24-s + 2.34·25-s − 2.61·26-s + 12.3·27-s + 1.51·28-s + ⋯ |
L(s) = 1 | − 1.21·2-s + 1.81·3-s + 0.469·4-s − 1.21·5-s − 2.20·6-s + 0.609·7-s + 0.643·8-s + 2.31·9-s + 1.46·10-s + 0.208·11-s + 0.854·12-s + 0.423·13-s − 0.738·14-s − 2.20·15-s − 1.24·16-s + 0.256·17-s − 2.80·18-s + 0.0294·19-s − 0.568·20-s + 1.10·21-s − 0.253·22-s + 0.392·23-s + 1.17·24-s + 0.468·25-s − 0.513·26-s + 2.38·27-s + 0.286·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.590744907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590744907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.692T + 11T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 - 0.128T + 19T^{2} \) |
| 23 | \( 1 - 1.88T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.17T + 59T^{2} \) |
| 61 | \( 1 - 2.63T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 6.49T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 - 1.20T + 97T^{2} \) |
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
−8.973168164071908477844587259229, −8.463450804264049008836011546390, −8.179229697067864261790785750173, −7.40019809579394625365892117437, −6.86386686395836697584981493611, −4.88380765499791948991774980387, −4.05042283675877981108185062968, −3.32256886059397489460219167165, −2.11455625839298733684141352946, −1.03327292513117171975883558716,
1.03327292513117171975883558716, 2.11455625839298733684141352946, 3.32256886059397489460219167165, 4.05042283675877981108185062968, 4.88380765499791948991774980387, 6.86386686395836697584981493611, 7.40019809579394625365892117437, 8.179229697067864261790785750173, 8.463450804264049008836011546390, 8.973168164071908477844587259229