| L(s) = 1 | − 1.32·2-s − 9.77·3-s − 6.25·4-s + 15.2·5-s + 12.9·6-s + 18.8·8-s + 68.4·9-s − 20.1·10-s + 11·11-s + 61.0·12-s + 28.1·13-s − 149.·15-s + 25.0·16-s + 6.48·17-s − 90.5·18-s + 69.4·19-s − 95.4·20-s − 14.5·22-s − 164.·23-s − 184.·24-s + 108.·25-s − 37.1·26-s − 405.·27-s + 172.·29-s + 197.·30-s − 52.1·31-s − 183.·32-s + ⋯ |
| L(s) = 1 | − 0.467·2-s − 1.88·3-s − 0.781·4-s + 1.36·5-s + 0.879·6-s + 0.832·8-s + 2.53·9-s − 0.638·10-s + 0.301·11-s + 1.46·12-s + 0.599·13-s − 2.56·15-s + 0.392·16-s + 0.0925·17-s − 1.18·18-s + 0.838·19-s − 1.06·20-s − 0.140·22-s − 1.49·23-s − 1.56·24-s + 0.864·25-s − 0.280·26-s − 2.88·27-s + 1.10·29-s + 1.20·30-s − 0.301·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8607188309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8607188309\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 1.32T + 8T^{2} \) |
| 3 | \( 1 + 9.77T + 27T^{2} \) |
| 5 | \( 1 - 15.2T + 125T^{2} \) |
| 13 | \( 1 - 28.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.48T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 52.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 126.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 70.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 210.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 732.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 110.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 560.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 484.T + 9.12e5T^{2} \) |
| 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
−10.24513332652502030691596335408, −9.860074334084390157940997835967, −8.950840688005675558722589270074, −7.59372689673138452046972502286, −6.43504199433984977457129896064, −5.77683838373788192100132223721, −5.09434951726220217882369619120, −4.04011624486354965679386525094, −1.66916963537411301076432959899, −0.72405847521537192560564841949,
0.72405847521537192560564841949, 1.66916963537411301076432959899, 4.04011624486354965679386525094, 5.09434951726220217882369619120, 5.77683838373788192100132223721, 6.43504199433984977457129896064, 7.59372689673138452046972502286, 8.950840688005675558722589270074, 9.860074334084390157940997835967, 10.24513332652502030691596335408
