| L(s) = 1 | + 1.77·2-s + 9·3-s − 28.8·4-s − 66.3·5-s + 15.9·6-s + 199.·7-s − 107.·8-s + 81·9-s − 117.·10-s + 239.·11-s − 259.·12-s − 611.·13-s + 352.·14-s − 597.·15-s + 732.·16-s + 143.·18-s − 2.15e3·19-s + 1.91e3·20-s + 1.79e3·21-s + 424.·22-s + 511.·23-s − 970.·24-s + 1.27e3·25-s − 1.08e3·26-s + 729·27-s − 5.74e3·28-s + 2.30e3·29-s + ⋯ |
| L(s) = 1 | + 0.313·2-s + 0.577·3-s − 0.901·4-s − 1.18·5-s + 0.180·6-s + 1.53·7-s − 0.595·8-s + 0.333·9-s − 0.371·10-s + 0.597·11-s − 0.520·12-s − 1.00·13-s + 0.480·14-s − 0.685·15-s + 0.715·16-s + 0.104·18-s − 1.36·19-s + 1.07·20-s + 0.886·21-s + 0.187·22-s + 0.201·23-s − 0.343·24-s + 0.409·25-s − 0.314·26-s + 0.192·27-s − 1.38·28-s + 0.508·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.77T + 32T^{2} \) |
| 5 | \( 1 + 66.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 199.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 239.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 611.T + 3.71e5T^{2} \) |
| 19 | \( 1 + 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 511.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.14e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.24e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.14e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.04e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 390.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.54e4T + 8.58e9T^{2} \) |
| 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
−8.675967818641984160999722610133, −8.223283411753450490114994447652, −7.64339398754239158297071663966, −6.46039373676219916876341491861, −4.92827972127091969004811585512, −4.52265659741420034647595756872, −3.81207741869404402660395378661, −2.56597594881698393388488224785, −1.21532807794349128656706961054, 0,
1.21532807794349128656706961054, 2.56597594881698393388488224785, 3.81207741869404402660395378661, 4.52265659741420034647595756872, 4.92827972127091969004811585512, 6.46039373676219916876341491861, 7.64339398754239158297071663966, 8.223283411753450490114994447652, 8.675967818641984160999722610133
