| L(s) = 1 | + 1.24·2-s − 0.445·4-s − 5-s + 0.246·7-s − 3.04·8-s − 1.24·10-s + 0.109·11-s + 0.307·14-s − 2.91·16-s + 1.50·17-s − 3.44·19-s + 0.445·20-s + 0.137·22-s + 7.80·23-s + 25-s − 0.109·28-s + 4.66·29-s − 3.27·31-s + 2.46·32-s + 1.87·34-s − 0.246·35-s − 0.939·37-s − 4.29·38-s + 3.04·40-s − 4.24·41-s − 11.2·43-s − 0.0489·44-s + ⋯ |
| L(s) = 1 | + 0.881·2-s − 0.222·4-s − 0.447·5-s + 0.0933·7-s − 1.07·8-s − 0.394·10-s + 0.0331·11-s + 0.0823·14-s − 0.727·16-s + 0.365·17-s − 0.790·19-s + 0.0995·20-s + 0.0292·22-s + 1.62·23-s + 0.200·25-s − 0.0207·28-s + 0.866·29-s − 0.588·31-s + 0.436·32-s + 0.322·34-s − 0.0417·35-s − 0.154·37-s − 0.696·38-s + 0.482·40-s − 0.663·41-s − 1.71·43-s − 0.00737·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012231462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012231462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 7 | \( 1 - 0.246T + 7T^{2} \) |
| 11 | \( 1 - 0.109T + 11T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 0.939T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 0.978T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 + 3.44T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 - 7.92T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
| 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
−8.024389345669544742541196567550, −6.83226419136044381349910396083, −6.59644929566442107370640607986, −5.54284256073700049769139658167, −4.92735182846864274769292958991, −4.48453573999484825895213978463, −3.43121062776999145237659948268, −3.16950463166251259910323172032, −1.92727172947874304862135295089, −0.61336708511041545822824695910,
0.61336708511041545822824695910, 1.92727172947874304862135295089, 3.16950463166251259910323172032, 3.43121062776999145237659948268, 4.48453573999484825895213978463, 4.92735182846864274769292958991, 5.54284256073700049769139658167, 6.59644929566442107370640607986, 6.83226419136044381349910396083, 8.024389345669544742541196567550
