L(s) = 1 | + 234·5-s + 729·9-s − 4.07e3·13-s − 990·17-s + 3.91e4·25-s − 3.18e4·29-s − 5.55e4·37-s − 8.49e4·41-s + 1.70e5·45-s + 1.17e5·49-s − 2.94e4·53-s + 2.34e5·61-s − 9.52e5·65-s + 4.27e5·73-s + 5.31e5·81-s − 2.31e5·85-s + 1.37e6·89-s − 1.47e6·97-s − 1.70e6·101-s + 4.58e5·109-s + 1.12e6·113-s − 2.96e6·117-s + ⋯ |
L(s) = 1 | + 1.87·5-s + 9-s − 1.85·13-s − 0.201·17-s + 2.50·25-s − 1.30·29-s − 1.09·37-s − 1.23·41-s + 1.87·45-s + 49-s − 0.197·53-s + 1.03·61-s − 3.46·65-s + 1.09·73-s + 81-s − 0.377·85-s + 1.95·89-s − 1.61·97-s − 1.65·101-s + 0.354·109-s + 0.777·113-s − 1.85·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.790086510\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790086510\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 - 234 T + p^{6} T^{2} \) |
| 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 + 4070 T + p^{6} T^{2} \) |
| 17 | \( 1 + 990 T + p^{6} T^{2} \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 + 31878 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( 1 + 55510 T + p^{6} T^{2} \) |
| 41 | \( 1 + 84942 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( 1 + 29430 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 - 234938 T + p^{6} T^{2} \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 427570 T + p^{6} T^{2} \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( 1 - 1378962 T + p^{6} T^{2} \) |
| 97 | \( 1 + 1472510 T + p^{6} T^{2} \) |
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
−17.66711598111247337833915300720, −16.79477526492107577068652082954, −14.92961960005412272282480914790, −13.66027329831071488364659636597, −12.54111176845652332787289646763, −10.27323120633818984044199962389, −9.426182773749689790760561697792, −6.96529283450080654709598085938, −5.19821066897336074860141084941, −2.03272536499668191245491348923,
2.03272536499668191245491348923, 5.19821066897336074860141084941, 6.96529283450080654709598085938, 9.426182773749689790760561697792, 10.27323120633818984044199962389, 12.54111176845652332787289646763, 13.66027329831071488364659636597, 14.92961960005412272282480914790, 16.79477526492107577068652082954, 17.66711598111247337833915300720