| L(s) = 1 | − 190·9-s + 1.53e3·11-s + 2.20e3·19-s + 1.12e4·29-s + 7.97e3·31-s + 3.08e3·41-s + 3.31e4·49-s + 5.67e4·59-s + 1.10e4·61-s − 8.47e4·71-s − 7.92e4·79-s − 2.29e4·81-s − 1.15e5·89-s − 2.91e5·99-s − 2.82e5·101-s − 4.36e5·109-s + 1.44e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.40e5·169-s + ⋯ |
| L(s) = 1 | − 0.781·9-s + 3.82·11-s + 1.39·19-s + 2.47·29-s + 1.49·31-s + 0.286·41-s + 1.97·49-s + 2.12·59-s + 0.380·61-s − 1.99·71-s − 1.42·79-s − 0.388·81-s − 1.54·89-s − 2.99·99-s − 2.75·101-s − 3.52·109-s + 8.98·121-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 + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.819563501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.819563501\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 190 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 33130 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 768 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 740470 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2696830 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1100 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8928490 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5610 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3988 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138667750 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1542 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 268756210 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 153278630 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 635715430 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 28380 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5522 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2088083650 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 42372 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1429023310 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 39640 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4298931010 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 57690 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3671481410 T^{2} + p^{10} T^{4} \) |
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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
−10.65335929466759936337545312410, −10.15893822531685608235511401134, −9.643235704591954400963775149991, −9.325860564398023733496074891392, −8.908934309594006724504838483060, −8.323952509192753902379838900065, −8.299169096156638168414030160296, −7.08865899247764819822239076232, −6.88104052576497074417768955433, −6.62868528995988956864764096511, −5.80388654690700414337193586015, −5.71887296143189412864602388035, −4.67479299495870732628045380727, −4.12686890211326358050869934821, −3.95211888331383763600277715954, −3.02358819192208971380518445172, −2.70757420039120224875917508315, −1.52034074498381282858868796696, −1.09470075451311396386672800053, −0.73900969042252917842222002760,
0.73900969042252917842222002760, 1.09470075451311396386672800053, 1.52034074498381282858868796696, 2.70757420039120224875917508315, 3.02358819192208971380518445172, 3.95211888331383763600277715954, 4.12686890211326358050869934821, 4.67479299495870732628045380727, 5.71887296143189412864602388035, 5.80388654690700414337193586015, 6.62868528995988956864764096511, 6.88104052576497074417768955433, 7.08865899247764819822239076232, 8.299169096156638168414030160296, 8.323952509192753902379838900065, 8.908934309594006724504838483060, 9.325860564398023733496074891392, 9.643235704591954400963775149991, 10.15893822531685608235511401134, 10.65335929466759936337545312410
