| L(s) = 1 | − 472·7-s − 243·9-s + 1.43e3·19-s + 6.25e3·25-s − 6.70e3·43-s + 1.33e5·49-s + 7.72e4·61-s + 1.14e5·63-s − 2.90e3·73-s + 5.90e4·81-s + 3.22e5·121-s + 127-s + 131-s − 6.75e5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.02e5·169-s − 3.47e5·171-s + 173-s − 2.95e6·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.64·7-s − 9-s + 0.910·19-s + 2·25-s − 0.552·43-s + 7.94·49-s + 2.65·61-s + 3.64·63-s − 0.0636·73-s + 81-s + 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s − 3.31·133-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.89·169-s − 0.910·171-s + 2.54e−6·173-s − 7.28·175-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8508468787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8508468787\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| 19 | $C_2$ | \( 1 - 1432 T + p^{5} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 236 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 1202 T + p^{5} T^{2} )( 1 + 1202 T + p^{5} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10324 T + p^{5} T^{2} )( 1 + 10324 T + p^{5} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 16550 T + p^{5} T^{2} )( 1 + 16550 T + p^{5} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 3352 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 38626 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 35536 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1450 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 100564 T + p^{5} T^{2} )( 1 + 100564 T + p^{5} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 134386 T + p^{5} T^{2} )( 1 + 134386 T + p^{5} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.67459109976696299341626958221, −11.10159692177237079436702051993, −10.37562045440413823429891023853, −10.15743375546035033567055426022, −9.474994045265618326136476083332, −9.430862831913607457721530748213, −8.722706731937565226370528518999, −8.400787569038800102951966720906, −7.23049959667702714916555790859, −7.05423797992535846440214891922, −6.44657812783019258670789780187, −6.13591241707981308590325422591, −5.55818079751533814993837864395, −4.91582781080138177169004617258, −3.62126655183978967047478476892, −3.58183533037601772624043169693, −2.78408885401117042355257541081, −2.59891091443881737168192794143, −0.895259633850279488718162203607, −0.34984497410458902256280642978,
0.34984497410458902256280642978, 0.895259633850279488718162203607, 2.59891091443881737168192794143, 2.78408885401117042355257541081, 3.58183533037601772624043169693, 3.62126655183978967047478476892, 4.91582781080138177169004617258, 5.55818079751533814993837864395, 6.13591241707981308590325422591, 6.44657812783019258670789780187, 7.05423797992535846440214891922, 7.23049959667702714916555790859, 8.400787569038800102951966720906, 8.722706731937565226370528518999, 9.430862831913607457721530748213, 9.474994045265618326136476083332, 10.15743375546035033567055426022, 10.37562045440413823429891023853, 11.10159692177237079436702051993, 11.67459109976696299341626958221
