| L(s) = 1 | + 2·2-s + 4·4-s − 10.4·5-s + 8·8-s − 20.8·10-s + 61.1·11-s − 59.2·13-s + 16·16-s + 20.4·17-s − 80.3·19-s − 41.7·20-s + 122.·22-s − 158.·23-s − 15.8·25-s − 118.·26-s − 85.1·29-s + 243.·31-s + 32·32-s + 40.9·34-s + 290.·37-s − 160.·38-s − 83.5·40-s − 168·41-s + 7.62·43-s + 244.·44-s − 316.·46-s − 169.·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.934·5-s + 0.353·8-s − 0.660·10-s + 1.67·11-s − 1.26·13-s + 0.250·16-s + 0.291·17-s − 0.970·19-s − 0.467·20-s + 1.18·22-s − 1.43·23-s − 0.127·25-s − 0.893·26-s − 0.545·29-s + 1.41·31-s + 0.176·32-s + 0.206·34-s + 1.29·37-s − 0.685·38-s − 0.330·40-s − 0.639·41-s + 0.0270·43-s + 0.837·44-s − 1.01·46-s − 0.525·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 10.4T + 125T^{2} \) |
| 11 | \( 1 - 61.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 85.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.62T + 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 805.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 33.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 277.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 768.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
| 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
−9.426656846252583507829530561966, −8.282405845033937796974126537100, −7.56561073402363566116041252355, −6.64412150997610083310075131701, −5.90666844954223240648343957521, −4.44027529517803970643855491784, −4.16256002873271091903360136028, −2.96751064038971173380436658843, −1.64559667598509747989666703614, 0,
1.64559667598509747989666703614, 2.96751064038971173380436658843, 4.16256002873271091903360136028, 4.44027529517803970643855491784, 5.90666844954223240648343957521, 6.64412150997610083310075131701, 7.56561073402363566116041252355, 8.282405845033937796974126537100, 9.426656846252583507829530561966
