| L(s) = 1 | + 3-s + 7-s + 6·13-s − 4·16-s + 2·19-s + 21-s − 3·25-s − 4·27-s + 6·31-s + 6·39-s + 18·41-s + 6·43-s − 4·48-s − 6·49-s + 2·57-s − 6·59-s − 3·75-s − 7·81-s − 6·89-s + 6·91-s + 6·93-s − 2·97-s − 8·103-s − 4·112-s − 13·121-s + 18·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1.66·13-s − 16-s + 0.458·19-s + 0.218·21-s − 3/5·25-s − 0.769·27-s + 1.07·31-s + 0.960·39-s + 2.81·41-s + 0.914·43-s − 0.577·48-s − 6/7·49-s + 0.264·57-s − 0.781·59-s − 0.346·75-s − 7/9·81-s − 0.635·89-s + 0.628·91-s + 0.622·93-s − 0.203·97-s − 0.788·103-s − 0.377·112-s − 1.18·121-s + 1.62·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53067 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53067 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807526880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807526880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.888061006803556127374571312211, −9.466522626635243878693684080639, −8.973164621931283772702834547338, −8.623356283001451932952991887560, −7.85525762032585291198822353618, −7.76717014448676428230485244695, −6.95290562753375386571004987690, −6.19927100600041456160946888575, −5.95653307947299162749031324311, −5.14580057535760759859548538628, −4.26584870437055125896430495982, −3.98006768240938057592965446342, −3.04379632027667628601308769871, −2.33466536702483286392915406569, −1.26126154367783016011002136790,
1.26126154367783016011002136790, 2.33466536702483286392915406569, 3.04379632027667628601308769871, 3.98006768240938057592965446342, 4.26584870437055125896430495982, 5.14580057535760759859548538628, 5.95653307947299162749031324311, 6.19927100600041456160946888575, 6.95290562753375386571004987690, 7.76717014448676428230485244695, 7.85525762032585291198822353618, 8.623356283001451932952991887560, 8.973164621931283772702834547338, 9.466522626635243878693684080639, 9.888061006803556127374571312211
