L(s) = 1 | + 5.17·2-s + 9·3-s − 5.19·4-s + 25·5-s + 46.5·6-s − 123.·7-s − 192.·8-s + 81·9-s + 129.·10-s − 121·11-s − 46.7·12-s − 500.·13-s − 639.·14-s + 225·15-s − 830.·16-s − 422.·17-s + 419.·18-s − 932.·19-s − 129.·20-s − 1.11e3·21-s − 626.·22-s − 1.22e3·23-s − 1.73e3·24-s + 625·25-s − 2.58e3·26-s + 729·27-s + 641.·28-s + ⋯ |
L(s) = 1 | + 0.915·2-s + 0.577·3-s − 0.162·4-s + 0.447·5-s + 0.528·6-s − 0.952·7-s − 1.06·8-s + 0.333·9-s + 0.409·10-s − 0.301·11-s − 0.0937·12-s − 0.820·13-s − 0.871·14-s + 0.258·15-s − 0.811·16-s − 0.354·17-s + 0.305·18-s − 0.592·19-s − 0.0726·20-s − 0.549·21-s − 0.275·22-s − 0.482·23-s − 0.614·24-s + 0.200·25-s − 0.751·26-s + 0.192·27-s + 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 5.17T + 32T^{2} \) |
| 7 | \( 1 + 123.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 500.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 422.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 932.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 159.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.81e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.15e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.15e4T + 8.58e9T^{2} \) |
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
−11.79625474022083768406984573646, −10.18133385577831479801716202034, −9.466133730767755497954006892179, −8.448841756819302632539310932998, −6.94196864888982131119391313733, −5.87617634421056300436678483836, −4.66526059050335581442195082675, −3.46188667418285481120055372018, −2.35840565968180940761034729307, 0,
2.35840565968180940761034729307, 3.46188667418285481120055372018, 4.66526059050335581442195082675, 5.87617634421056300436678483836, 6.94196864888982131119391313733, 8.448841756819302632539310932998, 9.466133730767755497954006892179, 10.18133385577831479801716202034, 11.79625474022083768406984573646