| L(s) = 1 | − 4·13-s − 12·23-s + 2·25-s + 4·37-s − 20·47-s + 12·49-s − 24·59-s − 20·61-s − 4·71-s − 20·73-s − 8·83-s + 24·97-s + 16·107-s + 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 2.50·23-s + 2/5·25-s + 0.657·37-s − 2.91·47-s + 12/7·49-s − 3.12·59-s − 2.56·61-s − 0.474·71-s − 2.34·73-s − 0.878·83-s + 2.43·97-s + 1.54·107-s + 1.91·109-s − 2·121-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−7.899616133536539659802552768606, −7.88323371610103274569630290427, −7.36676147311406525991899186062, −7.32744594519234089627938543196, −6.55380392175532091201362608242, −6.22431574515984770094900263234, −6.08115704914276999106653614760, −5.67157662749428763566210649157, −5.11964927604222898574508008028, −4.70231395392283951176946650693, −4.32544943824251300879595431885, −4.27417963083487563333039295376, −3.34712241476748950656995852028, −3.21115199665758050144526083509, −2.70923779164239140699052709052, −2.06832178346065980812863483855, −1.78586940765185142018552988697, −1.19792805662451518967251177407, 0, 0,
1.19792805662451518967251177407, 1.78586940765185142018552988697, 2.06832178346065980812863483855, 2.70923779164239140699052709052, 3.21115199665758050144526083509, 3.34712241476748950656995852028, 4.27417963083487563333039295376, 4.32544943824251300879595431885, 4.70231395392283951176946650693, 5.11964927604222898574508008028, 5.67157662749428763566210649157, 6.08115704914276999106653614760, 6.22431574515984770094900263234, 6.55380392175532091201362608242, 7.32744594519234089627938543196, 7.36676147311406525991899186062, 7.88323371610103274569630290427, 7.899616133536539659802552768606
