| L(s) = 1 | + 4·3-s + 4·5-s − 4·7-s + 8·9-s + 2·13-s + 16·15-s − 10·17-s + 8·19-s − 16·21-s − 4·23-s + 11·25-s + 12·27-s − 16·35-s − 2·37-s + 8·39-s − 12·43-s + 32·45-s + 4·47-s + 8·49-s − 40·51-s + 14·53-s + 32·57-s + 8·59-s + 8·61-s − 32·63-s + 8·65-s − 20·67-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s − 1.51·7-s + 8/3·9-s + 0.554·13-s + 4.13·15-s − 2.42·17-s + 1.83·19-s − 3.49·21-s − 0.834·23-s + 11/5·25-s + 2.30·27-s − 2.70·35-s − 0.328·37-s + 1.28·39-s − 1.82·43-s + 4.77·45-s + 0.583·47-s + 8/7·49-s − 5.60·51-s + 1.92·53-s + 4.23·57-s + 1.04·59-s + 1.02·61-s − 4.03·63-s + 0.992·65-s − 2.44·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.824858935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.824858935\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.76879059408955910240786623277, −11.57315681538910339168721901959, −10.40154355201536333201077454439, −10.34640463301166789617916303982, −9.779715327005286913427318400166, −9.524014131081033781687273619419, −8.933495585803860623297955552005, −8.648911364248645495779822676460, −8.623402503323118758598025292129, −7.55229769301900557870915055929, −6.97603214422178413439368945778, −6.74232386529797678794549029764, −6.02200135929139039453656588515, −5.61286764504178668945953674496, −4.69476694812651960367409068810, −3.91646776746271486704453338781, −3.28845776483147009444881669870, −2.75590623724216032251336061566, −2.36412307366741810259947605973, −1.59133414528139497970127095113,
1.59133414528139497970127095113, 2.36412307366741810259947605973, 2.75590623724216032251336061566, 3.28845776483147009444881669870, 3.91646776746271486704453338781, 4.69476694812651960367409068810, 5.61286764504178668945953674496, 6.02200135929139039453656588515, 6.74232386529797678794549029764, 6.97603214422178413439368945778, 7.55229769301900557870915055929, 8.623402503323118758598025292129, 8.648911364248645495779822676460, 8.933495585803860623297955552005, 9.524014131081033781687273619419, 9.779715327005286913427318400166, 10.34640463301166789617916303982, 10.40154355201536333201077454439, 11.57315681538910339168721901959, 11.76879059408955910240786623277
