| L(s) = 1 | − 2.42·2-s + 2.89·3-s + 3.87·4-s − 2.22·5-s − 7.02·6-s − 1.23·7-s − 4.54·8-s + 5.40·9-s + 5.39·10-s − 11-s + 11.2·12-s + 3.75·13-s + 2.98·14-s − 6.45·15-s + 3.26·16-s − 13.1·18-s + 3.89·19-s − 8.62·20-s − 3.56·21-s + 2.42·22-s − 8.98·23-s − 13.1·24-s − 0.0471·25-s − 9.09·26-s + 6.97·27-s − 4.77·28-s + 6.49·29-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.67·3-s + 1.93·4-s − 0.995·5-s − 2.86·6-s − 0.465·7-s − 1.60·8-s + 1.80·9-s + 1.70·10-s − 0.301·11-s + 3.24·12-s + 1.04·13-s + 0.797·14-s − 1.66·15-s + 0.817·16-s − 3.08·18-s + 0.894·19-s − 1.92·20-s − 0.779·21-s + 0.516·22-s − 1.87·23-s − 2.69·24-s − 0.00942·25-s − 1.78·26-s + 1.34·27-s − 0.901·28-s + 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188944476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188944476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 + 8.98T + 23T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 + 1.98T + 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 0.314T + 67T^{2} \) |
| 71 | \( 1 - 6.91T + 71T^{2} \) |
| 73 | \( 1 + 1.42T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 - 6.87T + 89T^{2} \) |
| 97 | \( 1 - 1.19T + 97T^{2} \) |
| 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
−8.397073860369279968953848634376, −8.106433801642357196871498868848, −7.82314812867509269710097009629, −6.87929504520883155256845150322, −6.15390139427147692858935450426, −4.42098075256741395274714843818, −3.56599402612317618961732755885, −2.87408165472402710125805831609, −1.94300729416617157298557488764, −0.78574335209616800920114889202,
0.78574335209616800920114889202, 1.94300729416617157298557488764, 2.87408165472402710125805831609, 3.56599402612317618961732755885, 4.42098075256741395274714843818, 6.15390139427147692858935450426, 6.87929504520883155256845150322, 7.82314812867509269710097009629, 8.106433801642357196871498868848, 8.397073860369279968953848634376
