| L(s) = 1 | − 4·7-s − 8·17-s − 16·23-s + 6·25-s − 12·31-s + 24·41-s + 16·47-s − 2·49-s − 4·73-s − 28·79-s − 16·89-s − 4·97-s − 28·103-s − 16·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.94·17-s − 3.33·23-s + 6/5·25-s − 2.15·31-s + 3.74·41-s + 2.33·47-s − 2/7·49-s − 0.468·73-s − 3.15·79-s − 1.69·89-s − 0.406·97-s − 2.75·103-s − 1.50·113-s + 2.93·119-s + 6/11·121-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 + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.954994220680132586444216566726, −8.523279014789570155655082981833, −7.974876569952133207354443364672, −7.70801649405573358764580996639, −7.09703546460804970094605696028, −6.93987392654325592399589061998, −6.52194513518395794877911625388, −5.92330826315976788225337408832, −5.76321471351560276080619964280, −5.62211798756695795233772317844, −4.47665227301461539913999542338, −4.38989350882677036367859044587, −3.91997908796352962291567467270, −3.64893927339412581488796471968, −2.66407365747891032890302225959, −2.64418903952155888491435109524, −2.04847029873342690456101871207, −1.26543081489204407561362237186, 0, 0,
1.26543081489204407561362237186, 2.04847029873342690456101871207, 2.64418903952155888491435109524, 2.66407365747891032890302225959, 3.64893927339412581488796471968, 3.91997908796352962291567467270, 4.38989350882677036367859044587, 4.47665227301461539913999542338, 5.62211798756695795233772317844, 5.76321471351560276080619964280, 5.92330826315976788225337408832, 6.52194513518395794877911625388, 6.93987392654325592399589061998, 7.09703546460804970094605696028, 7.70801649405573358764580996639, 7.974876569952133207354443364672, 8.523279014789570155655082981833, 8.954994220680132586444216566726
