L(s) = 1 | + 5·9-s − 2·11-s − 16·19-s + 10·31-s + 10·49-s − 6·59-s − 8·61-s + 30·71-s − 4·79-s + 16·81-s + 18·89-s − 10·99-s + 36·101-s − 4·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 80·171-s + 173-s + ⋯ |
L(s) = 1 | + 5/3·9-s − 0.603·11-s − 3.67·19-s + 1.79·31-s + 10/7·49-s − 0.781·59-s − 1.02·61-s + 3.56·71-s − 0.450·79-s + 16/9·81-s + 1.90·89-s − 1.00·99-s + 3.58·101-s − 0.383·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 6.11·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.865468372\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.865468372\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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
−10.01313492178329116891562054435, −9.940319271789387594922043631822, −9.051961255465652894133445667175, −8.946503632090740994143460853420, −8.351345440658039369721313226047, −8.062373686194712659345625370078, −7.55982586091152438721473719724, −7.21803553558152379474616634641, −6.44007201054515425127993892638, −6.43930632135198331221780231950, −6.13009207437561717601615796590, −5.17527300427536703996650957480, −4.71045373961920044703539823180, −4.49530473247167458712421475269, −3.93062226249658557115565219678, −3.58388138069755241678952823937, −2.38937898927773957860052164305, −2.37997536664755934843867088895, −1.58400607519580161411540482929, −0.61015075114485358519503579589,
0.61015075114485358519503579589, 1.58400607519580161411540482929, 2.37997536664755934843867088895, 2.38937898927773957860052164305, 3.58388138069755241678952823937, 3.93062226249658557115565219678, 4.49530473247167458712421475269, 4.71045373961920044703539823180, 5.17527300427536703996650957480, 6.13009207437561717601615796590, 6.43930632135198331221780231950, 6.44007201054515425127993892638, 7.21803553558152379474616634641, 7.55982586091152438721473719724, 8.062373686194712659345625370078, 8.351345440658039369721313226047, 8.946503632090740994143460853420, 9.051961255465652894133445667175, 9.940319271789387594922043631822, 10.01313492178329116891562054435