L(s) = 1 | + 9-s + 8·11-s − 4·13-s − 8·19-s − 8·23-s − 6·25-s + 12·29-s − 12·41-s + 8·43-s − 14·49-s − 8·67-s + 20·73-s − 16·79-s + 81-s − 8·83-s + 8·99-s − 36·101-s + 32·103-s − 24·107-s − 4·117-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + ⋯ |
L(s) = 1 | + 1/3·9-s + 2.41·11-s − 1.10·13-s − 1.83·19-s − 1.66·23-s − 6/5·25-s + 2.22·29-s − 1.87·41-s + 1.21·43-s − 2·49-s − 0.977·67-s + 2.34·73-s − 1.80·79-s + 1/9·81-s − 0.878·83-s + 0.804·99-s − 3.58·101-s + 3.15·103-s − 2.32·107-s − 0.369·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.098990694093691505068092710868, −7.937468336979128641342085392290, −7.11481053133502324948838023953, −6.68843767199302438025917244953, −6.42897107072744719896577758454, −6.14665377703923964815323451966, −5.42397128942512527640781722495, −4.69316359438713629341208168682, −4.25303028692796488061253338187, −4.06945576769544587749203909991, −3.40732011658555077078177858924, −2.56114706993332614075312120102, −1.90162997266294366276999906239, −1.37597014489453523765395251943, 0,
1.37597014489453523765395251943, 1.90162997266294366276999906239, 2.56114706993332614075312120102, 3.40732011658555077078177858924, 4.06945576769544587749203909991, 4.25303028692796488061253338187, 4.69316359438713629341208168682, 5.42397128942512527640781722495, 6.14665377703923964815323451966, 6.42897107072744719896577758454, 6.68843767199302438025917244953, 7.11481053133502324948838023953, 7.937468336979128641342085392290, 8.098990694093691505068092710868