| L(s) = 1 | − 16·4-s − 264·11-s + 256·16-s − 1.00e3·19-s + 4.38e3·29-s + 4.62e3·31-s − 2.48e3·41-s + 4.22e3·44-s + 4.03e3·49-s + 1.59e4·59-s + 3.32e4·61-s − 4.09e3·64-s + 4.90e4·71-s + 1.60e4·76-s + 9.24e4·79-s − 2.20e5·89-s − 2.83e5·101-s + 3.49e5·109-s − 7.00e4·116-s − 2.69e5·121-s − 7.39e4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.657·11-s + 1/4·16-s − 0.635·19-s + 0.967·29-s + 0.864·31-s − 0.230·41-s + 0.328·44-s + 0.239·49-s + 0.596·59-s + 1.14·61-s − 1/8·64-s + 1.15·71-s + 0.317·76-s + 1.66·79-s − 2.95·89-s − 2.76·101-s + 2.81·109-s − 0.483·116-s − 1.67·121-s − 0.432·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.097730727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097730727\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 4030 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 p T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 152330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2790430 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 500 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 170590 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2190 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2312 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 12305350 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1242 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 131332490 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 415288270 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 392614630 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7980 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 16622 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2696981350 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 24528 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3726958510 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 46240 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 5217997510 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 110310 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11030942590 T^{2} + p^{10} T^{4} \) |
| 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
−10.38858643790891307381281046454, −10.11697058626191480572388079006, −9.600768136078730718718491891313, −9.270619998556645550580208910379, −8.504965912789616761636397457195, −8.273541276897703962152377969164, −8.046282784532476610438762097517, −7.23705098257788658970033089901, −6.78512394185472005897638334966, −6.42387844313989723250318154235, −5.53723246327256231871992359949, −5.48388134618057891145764387822, −4.60139646379683965057252500997, −4.38369306649188489267908197685, −3.67157197122802581019873530613, −3.01968227939375626281256026143, −2.46942471888206293635502685017, −1.81754458528218558292932608072, −0.911619783185712865114913543018, −0.43007168670062072096296076288,
0.43007168670062072096296076288, 0.911619783185712865114913543018, 1.81754458528218558292932608072, 2.46942471888206293635502685017, 3.01968227939375626281256026143, 3.67157197122802581019873530613, 4.38369306649188489267908197685, 4.60139646379683965057252500997, 5.48388134618057891145764387822, 5.53723246327256231871992359949, 6.42387844313989723250318154235, 6.78512394185472005897638334966, 7.23705098257788658970033089901, 8.046282784532476610438762097517, 8.273541276897703962152377969164, 8.504965912789616761636397457195, 9.270619998556645550580208910379, 9.600768136078730718718491891313, 10.11697058626191480572388079006, 10.38858643790891307381281046454
