| L(s) = 1 | − 81·9-s + 92·19-s + 388·31-s − 4.03e3·49-s − 3.93e3·61-s − 1.53e4·79-s + 6.56e3·81-s − 4.40e4·109-s + 2.92e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.58e4·169-s − 7.45e3·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s + 0.254·19-s + 0.403·31-s − 1.68·49-s − 1.05·61-s − 2.46·79-s + 81-s − 3.70·109-s + 2·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 1.25·169-s − 0.254·171-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.283028890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283028890\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 4034 T^{2} + p^{8} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 35806 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 46 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 194 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 503522 T^{2} + p^{8} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3492194 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1966 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 5421406 T^{2} + p^{8} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 16169282 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7682 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 176908034 T^{2} + p^{8} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.80186086171773889268757221310, −10.70195456172704281703235907155, −10.68463760908044173563462002749, −9.751157445268544823063159032697, −9.548279674062075663627545709271, −8.990691709066853607017936745048, −8.318378357536452688311592724548, −8.190970242510710549382918987095, −7.52511937235404200572035669819, −6.88614149310971935584661024164, −6.47453403141592055500578710885, −5.71053272850808964196648925890, −5.54380542210282768278335034907, −4.67019824087582458927950128806, −4.26867895857641914014317401282, −3.24853304718201481237950096378, −3.02153227872595687725641750912, −2.12849548509117009807092110044, −1.33083883859862065812651018751, −0.35823566525021622926176344114,
0.35823566525021622926176344114, 1.33083883859862065812651018751, 2.12849548509117009807092110044, 3.02153227872595687725641750912, 3.24853304718201481237950096378, 4.26867895857641914014317401282, 4.67019824087582458927950128806, 5.54380542210282768278335034907, 5.71053272850808964196648925890, 6.47453403141592055500578710885, 6.88614149310971935584661024164, 7.52511937235404200572035669819, 8.190970242510710549382918987095, 8.318378357536452688311592724548, 8.990691709066853607017936745048, 9.548279674062075663627545709271, 9.751157445268544823063159032697, 10.68463760908044173563462002749, 10.70195456172704281703235907155, 11.80186086171773889268757221310
