| L(s) = 1 | − 2-s + 3-s − 6-s + 2·7-s + 8-s − 6·11-s + 7·13-s − 2·14-s − 16-s − 3·17-s − 2·19-s + 2·21-s + 6·22-s − 6·23-s + 24-s − 7·26-s − 27-s − 3·29-s − 8·31-s − 6·33-s + 3·34-s − 7·37-s + 2·38-s + 7·39-s + 3·41-s − 2·42-s − 10·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.755·7-s + 0.353·8-s − 1.80·11-s + 1.94·13-s − 0.534·14-s − 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.436·21-s + 1.27·22-s − 1.25·23-s + 0.204·24-s − 1.37·26-s − 0.192·27-s − 0.557·29-s − 1.43·31-s − 1.04·33-s + 0.514·34-s − 1.15·37-s + 0.324·38-s + 1.12·39-s + 0.468·41-s − 0.308·42-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7110909140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7110909140\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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
−9.296293690340900925765819938554, −8.715927878463112715413151348196, −8.643717069149459106011447601934, −8.274442127281453992446882320501, −7.921729590370679079173151828655, −7.74853050479598642576672807887, −7.19750935368901669164601888926, −6.61878459451499097519724862605, −6.37676094692263901996284236351, −5.62554748738814150318699852792, −5.48689662705316968474346515922, −4.96379307788970275424519328362, −4.49192324638086014736300277949, −3.76051256825885755036106234753, −3.71382677890955821401430475414, −2.98608368215299611266035932527, −2.36785093839935018985393230343, −1.73483325226765150085499433309, −1.60552103111014607058052376755, −0.31905442401513903992629206416,
0.31905442401513903992629206416, 1.60552103111014607058052376755, 1.73483325226765150085499433309, 2.36785093839935018985393230343, 2.98608368215299611266035932527, 3.71382677890955821401430475414, 3.76051256825885755036106234753, 4.49192324638086014736300277949, 4.96379307788970275424519328362, 5.48689662705316968474346515922, 5.62554748738814150318699852792, 6.37676094692263901996284236351, 6.61878459451499097519724862605, 7.19750935368901669164601888926, 7.74853050479598642576672807887, 7.921729590370679079173151828655, 8.274442127281453992446882320501, 8.643717069149459106011447601934, 8.715927878463112715413151348196, 9.296293690340900925765819938554
