L(s) = 1 | + 2·3-s + 3·9-s − 2·17-s − 8·19-s + 6·25-s + 4·27-s − 20·41-s + 8·43-s − 10·49-s − 4·51-s − 16·57-s − 24·59-s + 24·67-s + 4·73-s + 12·75-s + 5·81-s + 24·83-s − 4·89-s + 12·97-s + 4·113-s − 22·121-s − 40·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 0.485·17-s − 1.83·19-s + 6/5·25-s + 0.769·27-s − 3.12·41-s + 1.21·43-s − 1.42·49-s − 0.560·51-s − 2.11·57-s − 3.12·59-s + 2.93·67-s + 0.468·73-s + 1.38·75-s + 5/9·81-s + 2.63·83-s − 0.423·89-s + 1.21·97-s + 0.376·113-s − 2·121-s − 3.60·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.56738111269202357438702690211, −6.81173000092255075251437332159, −6.63257438834160544258861284685, −6.46249354410612141225423166895, −5.77904204714328784200418377249, −4.91997035368174263783342884746, −4.91585848792345990418783522770, −4.36159048011225245066599957337, −3.75890531283655162147766509504, −3.36779318987556938646346405293, −2.94610759542800375463162022317, −2.09209863939800055422786202913, −2.07543878259651085165424781495, −1.16529472267103972455357580300, 0,
1.16529472267103972455357580300, 2.07543878259651085165424781495, 2.09209863939800055422786202913, 2.94610759542800375463162022317, 3.36779318987556938646346405293, 3.75890531283655162147766509504, 4.36159048011225245066599957337, 4.91585848792345990418783522770, 4.91997035368174263783342884746, 5.77904204714328784200418377249, 6.46249354410612141225423166895, 6.63257438834160544258861284685, 6.81173000092255075251437332159, 7.56738111269202357438702690211