| L(s) = 1 | + 2·3-s + 3·9-s + 2·17-s − 6·25-s + 4·27-s − 12·41-s − 8·43-s + 10·49-s + 4·51-s − 16·59-s + 4·73-s − 12·75-s + 5·81-s − 16·83-s + 4·89-s − 4·97-s − 4·113-s − 6·121-s − 24·123-s + 127-s − 16·129-s + 131-s + 137-s + 139-s + 20·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.485·17-s − 6/5·25-s + 0.769·27-s − 1.87·41-s − 1.21·43-s + 10/7·49-s + 0.560·51-s − 2.08·59-s + 0.468·73-s − 1.38·75-s + 5/9·81-s − 1.75·83-s + 0.423·89-s − 0.406·97-s − 0.376·113-s − 0.545·121-s − 2.16·123-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.64·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.52493107867361885385949519610, −7.12123359751751697564522926438, −6.63294217184781215622607650009, −6.23298293041250704903404966164, −5.71978715523291227280017627311, −5.20625892855017883150777120937, −4.83145784884060780585774379641, −4.17145127026144677670871079526, −3.87977293758958657482788696695, −3.28108709549526633078410035224, −3.00493005944219356696117765282, −2.30983338487595644475607398916, −1.78004437277351556253637931229, −1.25659079395901986265187908802, 0,
1.25659079395901986265187908802, 1.78004437277351556253637931229, 2.30983338487595644475607398916, 3.00493005944219356696117765282, 3.28108709549526633078410035224, 3.87977293758958657482788696695, 4.17145127026144677670871079526, 4.83145784884060780585774379641, 5.20625892855017883150777120937, 5.71978715523291227280017627311, 6.23298293041250704903404966164, 6.63294217184781215622607650009, 7.12123359751751697564522926438, 7.52493107867361885385949519610
