| L(s) = 1 | − 20·4-s − 504·11-s − 624·16-s + 440·19-s − 1.38e4·29-s + 1.35e4·31-s + 396·41-s + 1.00e4·44-s − 3.00e4·49-s − 4.93e4·59-s − 1.13e4·61-s + 3.29e4·64-s − 1.06e5·71-s − 8.80e3·76-s − 1.03e5·79-s − 1.99e4·89-s + 2.18e5·101-s + 4.20e4·109-s + 2.77e5·116-s − 1.31e5·121-s − 2.70e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 5/8·4-s − 1.25·11-s − 0.609·16-s + 0.279·19-s − 3.06·29-s + 2.52·31-s + 0.0367·41-s + 0.784·44-s − 1.78·49-s − 1.84·59-s − 0.392·61-s + 1.00·64-s − 2.51·71-s − 0.174·76-s − 1.87·79-s − 0.267·89-s + 2.12·101-s + 0.338·109-s + 1.91·116-s − 0.817·121-s − 1.57·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 5 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 30050 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 252 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 728330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2363810 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6946370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6930 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6752 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 56462470 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 198 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 293842250 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 347593490 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 802472090 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 795787610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 53352 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 883886830 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4053674810 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6923133890 T^{2} + p^{10} T^{4} \) |
| 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
−11.21099699103976252258243336241, −10.62441995446558768900099680517, −10.03913451677762035148717278142, −9.788242869420337006876097678141, −9.037649110502722844995713759263, −8.869928944894626396329292624242, −7.985718088998878583965738641626, −7.78880383775668252870374696115, −7.22106824889611419125741888276, −6.49224615672729731359790224420, −5.86013416013236333928805948267, −5.41191053299819393026858529180, −4.57974691383914191498075178471, −4.48529443556198924561564843327, −3.42636308197417383774331698870, −2.89023478441739077057282056318, −2.09127591660887759203130553733, −1.27520153306779139054072249281, 0, 0,
1.27520153306779139054072249281, 2.09127591660887759203130553733, 2.89023478441739077057282056318, 3.42636308197417383774331698870, 4.48529443556198924561564843327, 4.57974691383914191498075178471, 5.41191053299819393026858529180, 5.86013416013236333928805948267, 6.49224615672729731359790224420, 7.22106824889611419125741888276, 7.78880383775668252870374696115, 7.985718088998878583965738641626, 8.869928944894626396329292624242, 9.037649110502722844995713759263, 9.788242869420337006876097678141, 10.03913451677762035148717278142, 10.62441995446558768900099680517, 11.21099699103976252258243336241
