| L(s) = 1 | − 2·3-s + 4-s + 3·9-s − 2·12-s + 6·13-s + 16-s + 4·17-s + 16·23-s − 6·25-s − 4·27-s − 4·29-s + 3·36-s − 12·39-s − 8·43-s − 2·48-s + 49-s − 8·51-s + 6·52-s + 12·53-s + 12·61-s + 64-s + 4·68-s − 32·69-s + 12·75-s + 5·81-s + 8·87-s + 16·92-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 9-s − 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.970·17-s + 3.33·23-s − 6/5·25-s − 0.769·27-s − 0.742·29-s + 1/2·36-s − 1.92·39-s − 1.21·43-s − 0.288·48-s + 1/7·49-s − 1.12·51-s + 0.832·52-s + 1.64·53-s + 1.53·61-s + 1/8·64-s + 0.485·68-s − 3.85·69-s + 1.38·75-s + 5/9·81-s + 0.857·87-s + 1.66·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675019172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675019172\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.964617363597458662832720006765, −8.356918966632091922574869118017, −7.78249907504316350377739128427, −7.30413306038065925571803859155, −6.84552480602986516155667793601, −6.50995416203766137108906926807, −5.94122016189281733370585672798, −5.33985014787602837985094112170, −5.31567853187232470908018483085, −4.47695643349088345789887691529, −3.62482887081886485478101246558, −3.47145333357673570260660471710, −2.49738125642812316298615788516, −1.45112495973791681767006881356, −0.926744370121951468387320874015,
0.926744370121951468387320874015, 1.45112495973791681767006881356, 2.49738125642812316298615788516, 3.47145333357673570260660471710, 3.62482887081886485478101246558, 4.47695643349088345789887691529, 5.31567853187232470908018483085, 5.33985014787602837985094112170, 5.94122016189281733370585672798, 6.50995416203766137108906926807, 6.84552480602986516155667793601, 7.30413306038065925571803859155, 7.78249907504316350377739128427, 8.356918966632091922574869118017, 8.964617363597458662832720006765
