| L(s) = 1 | − 2·3-s − 5-s + 2·7-s + 3·9-s + 6·11-s − 13-s + 2·15-s + 4·19-s − 4·21-s + 2·23-s − 8·25-s − 4·27-s − 8·29-s + 6·31-s − 12·33-s − 2·35-s + 10·37-s + 2·39-s − 6·41-s + 7·43-s − 3·45-s + 14·47-s + 3·49-s − 3·53-s − 6·55-s − 8·57-s + 17·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.755·7-s + 9-s + 1.80·11-s − 0.277·13-s + 0.516·15-s + 0.917·19-s − 0.872·21-s + 0.417·23-s − 8/5·25-s − 0.769·27-s − 1.48·29-s + 1.07·31-s − 2.08·33-s − 0.338·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.06·43-s − 0.447·45-s + 2.04·47-s + 3/7·49-s − 0.412·53-s − 0.809·55-s − 1.05·57-s + 2.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.908043301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.908043301\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 189 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 117 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 153 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 157 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−7.81757599108416002108208553075, −7.76510697465294518208095151697, −7.24668686245608572676297219072, −7.03215543771166556290692087471, −6.50136686595623507719007010167, −6.48671595727903152365567502304, −5.79502308844506216933928996637, −5.67497380672793220589644645072, −5.18604481537598688761659513029, −5.10040932924411161093909520212, −4.29294202007692834825117800997, −4.18090917364503751735920203824, −3.80084775561380021621834420549, −3.73743748736139043819899988461, −2.80872153513425669464973692142, −2.44175164493076720499479069033, −1.82154885748670611987139882791, −1.46034976645961276655715718156, −0.77591689989313673412833269959, −0.64602063457949569871679077669,
0.64602063457949569871679077669, 0.77591689989313673412833269959, 1.46034976645961276655715718156, 1.82154885748670611987139882791, 2.44175164493076720499479069033, 2.80872153513425669464973692142, 3.73743748736139043819899988461, 3.80084775561380021621834420549, 4.18090917364503751735920203824, 4.29294202007692834825117800997, 5.10040932924411161093909520212, 5.18604481537598688761659513029, 5.67497380672793220589644645072, 5.79502308844506216933928996637, 6.48671595727903152365567502304, 6.50136686595623507719007010167, 7.03215543771166556290692087471, 7.24668686245608572676297219072, 7.76510697465294518208095151697, 7.81757599108416002108208553075
