L(s) = 1 | + 6·7-s + 2·9-s + 2·11-s + 25-s + 12·37-s − 6·43-s + 14·49-s + 12·63-s + 12·77-s − 5·81-s − 6·83-s − 12·89-s − 8·97-s + 4·99-s + 14·107-s − 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯ |
L(s) = 1 | + 2.26·7-s + 2/3·9-s + 0.603·11-s + 1/5·25-s + 1.97·37-s − 0.914·43-s + 2·49-s + 1.51·63-s + 1.36·77-s − 5/9·81-s − 0.658·83-s − 1.27·89-s − 0.812·97-s + 0.402·99-s + 1.35·107-s − 0.636·121-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 + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.905513878\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.905513878\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.485164576854354017204415795327, −8.265894433756271009618218954782, −7.81206781165186907426945019783, −7.33830605375437367174840694811, −6.97785035415139349026961231337, −6.32339970111700000853005075332, −5.80129611978238869113739161971, −5.24625666566422344934059906548, −4.67598662920642177027627521559, −4.44874675827745828161559199886, −3.94168404816196990425525905487, −3.09797412820697803496565581152, −2.26418008765287970254190043192, −1.61669382345785442524926174877, −1.10302363199758985570479289802,
1.10302363199758985570479289802, 1.61669382345785442524926174877, 2.26418008765287970254190043192, 3.09797412820697803496565581152, 3.94168404816196990425525905487, 4.44874675827745828161559199886, 4.67598662920642177027627521559, 5.24625666566422344934059906548, 5.80129611978238869113739161971, 6.32339970111700000853005075332, 6.97785035415139349026961231337, 7.33830605375437367174840694811, 7.81206781165186907426945019783, 8.265894433756271009618218954782, 8.485164576854354017204415795327