| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s − 5·7-s + 4·8-s − 2·10-s + 11-s − 3·12-s − 4·13-s − 10·14-s + 15-s + 5·16-s − 4·17-s − 4·19-s − 3·20-s + 5·21-s + 2·22-s + 4·23-s − 4·24-s − 8·26-s + 27-s − 15·28-s − 18·29-s + 2·30-s − 11·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s − 1.88·7-s + 1.41·8-s − 0.632·10-s + 0.301·11-s − 0.866·12-s − 1.10·13-s − 2.67·14-s + 0.258·15-s + 5/4·16-s − 0.970·17-s − 0.917·19-s − 0.670·20-s + 1.09·21-s + 0.426·22-s + 0.834·23-s − 0.816·24-s − 1.56·26-s + 0.192·27-s − 2.83·28-s − 3.34·29-s + 0.365·30-s − 1.97·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171845662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171845662\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 31 | $C_2$ | \( 1 + 11 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 4 T - 55 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.44059175555148158650831935479, −9.857504565905466463292229302038, −9.564701528093817623249267999424, −8.990929058070490281775503587306, −8.886089635943010266331329871292, −7.933777377430778476396034650552, −7.43737736408117011508053811949, −7.04511256942174932746265243249, −6.70767701792017356476275563227, −6.58137146291907445886427305690, −5.71234934582740554246669523891, −5.43557313652231633360915989398, −5.31462483362516590380693252744, −4.20071517198686891558781611059, −4.16317205055884609459573648081, −3.49955072810898198055494204159, −3.23778548332665826473118592906, −2.18595838288899145473078484774, −2.13496396510462275707293512527, −0.38548791886378029315999885008,
0.38548791886378029315999885008, 2.13496396510462275707293512527, 2.18595838288899145473078484774, 3.23778548332665826473118592906, 3.49955072810898198055494204159, 4.16317205055884609459573648081, 4.20071517198686891558781611059, 5.31462483362516590380693252744, 5.43557313652231633360915989398, 5.71234934582740554246669523891, 6.58137146291907445886427305690, 6.70767701792017356476275563227, 7.04511256942174932746265243249, 7.43737736408117011508053811949, 7.933777377430778476396034650552, 8.886089635943010266331329871292, 8.990929058070490281775503587306, 9.564701528093817623249267999424, 9.857504565905466463292229302038, 10.44059175555148158650831935479
