| L(s) = 1 | − 2-s − 3·3-s − 7·4-s + 12·5-s + 3·6-s + 15·8-s + 9·9-s − 12·10-s + 20·11-s + 21·12-s − 84·13-s − 36·15-s + 41·16-s − 96·17-s − 9·18-s + 12·19-s − 84·20-s − 20·22-s − 176·23-s − 45·24-s + 19·25-s + 84·26-s − 27·27-s + 58·29-s + 36·30-s − 264·31-s − 161·32-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 1.07·5-s + 0.204·6-s + 0.662·8-s + 1/3·9-s − 0.379·10-s + 0.548·11-s + 0.505·12-s − 1.79·13-s − 0.619·15-s + 0.640·16-s − 1.36·17-s − 0.117·18-s + 0.144·19-s − 0.939·20-s − 0.193·22-s − 1.59·23-s − 0.382·24-s + 0.151·25-s + 0.633·26-s − 0.192·27-s + 0.371·29-s + 0.219·30-s − 1.52·31-s − 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 84 T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 - 12 T + p^{3} T^{2} \) |
| 23 | \( 1 + 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 264 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 156 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 722 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 492 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 240 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 - 924 T + p^{3} T^{2} \) |
| 89 | \( 1 + 744 T + p^{3} T^{2} \) |
| 97 | \( 1 + 168 T + p^{3} T^{2} \) |
| 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
−12.20195726756650422241073522087, −10.88007903490593513500051988762, −9.672135827053330853225560735685, −9.454514758526488831897198280441, −7.903931117775059840010464044635, −6.56029385957782858486388932900, −5.35999822990028879429235403923, −4.32594732984773623637445150922, −1.98366565515813513033987205203, 0,
1.98366565515813513033987205203, 4.32594732984773623637445150922, 5.35999822990028879429235403923, 6.56029385957782858486388932900, 7.903931117775059840010464044635, 9.454514758526488831897198280441, 9.672135827053330853225560735685, 10.88007903490593513500051988762, 12.20195726756650422241073522087
