L(s) = 1 | − 5.88·2-s + 9·3-s + 2.57·4-s + 25·5-s − 52.9·6-s − 219.·7-s + 173.·8-s + 81·9-s − 147.·10-s + 121·11-s + 23.1·12-s − 842.·13-s + 1.28e3·14-s + 225·15-s − 1.09e3·16-s + 1.32e3·17-s − 476.·18-s + 2.03e3·19-s + 64.4·20-s − 1.97e3·21-s − 711.·22-s − 2.82e3·23-s + 1.55e3·24-s + 625·25-s + 4.95e3·26-s + 729·27-s − 564.·28-s + ⋯ |
L(s) = 1 | − 1.03·2-s + 0.577·3-s + 0.0805·4-s + 0.447·5-s − 0.600·6-s − 1.69·7-s + 0.955·8-s + 0.333·9-s − 0.464·10-s + 0.301·11-s + 0.0465·12-s − 1.38·13-s + 1.75·14-s + 0.258·15-s − 1.07·16-s + 1.11·17-s − 0.346·18-s + 1.29·19-s + 0.0360·20-s − 0.976·21-s − 0.313·22-s − 1.11·23-s + 0.551·24-s + 0.200·25-s + 1.43·26-s + 0.192·27-s − 0.136·28-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
\(L(3)\) |
\(\approx\) |
\(0.9814060019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9814060019\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.88T + 32T^{2} \) |
| 7 | \( 1 + 219.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 842.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.94e4T + 8.58e9T^{2} \) |
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
−11.99339477945811330027777975154, −10.23616630353458251510032546857, −9.640998044798032636588912381242, −9.316038365928505487052159813226, −7.83058088679783850096457909883, −7.05252906335094775373156270013, −5.59122103491662063093064932843, −3.81595058571337851112764898732, −2.44722061012772049088955662139, −0.71719252967250954768286549215,
0.71719252967250954768286549215, 2.44722061012772049088955662139, 3.81595058571337851112764898732, 5.59122103491662063093064932843, 7.05252906335094775373156270013, 7.83058088679783850096457909883, 9.316038365928505487052159813226, 9.640998044798032636588912381242, 10.23616630353458251510032546857, 11.99339477945811330027777975154