| L(s) = 1 | − 3-s + 4-s − 2·9-s − 12-s + 4·13-s − 3·16-s − 3·17-s − 12·23-s − 2·25-s + 5·27-s − 15·29-s − 2·36-s − 4·39-s − 5·43-s + 3·48-s + 49-s + 3·51-s + 4·52-s + 3·53-s + 5·61-s − 7·64-s − 3·68-s + 12·69-s + 2·75-s + 11·79-s + 81-s + 15·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 2/3·9-s − 0.288·12-s + 1.10·13-s − 3/4·16-s − 0.727·17-s − 2.50·23-s − 2/5·25-s + 0.962·27-s − 2.78·29-s − 1/3·36-s − 0.640·39-s − 0.762·43-s + 0.433·48-s + 1/7·49-s + 0.420·51-s + 0.554·52-s + 0.412·53-s + 0.640·61-s − 7/8·64-s − 0.363·68-s + 1.44·69-s + 0.230·75-s + 1.23·79-s + 1/9·81-s + 1.60·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
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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
−9.513325163360725882556033794995, −9.077417084015230940122177295816, −8.487995514415829589315400073807, −8.068071510872389098162573713865, −7.49744079817246587850200802404, −6.85017343928106467007064060318, −6.34428311741507214172984550860, −5.84635288065204829999376709079, −5.59011460426287360473884640673, −4.70074891919111595018573788134, −3.92114272928466676617818582957, −3.54616840774244918324171029601, −2.34337929206565910714894249260, −1.83194551960988828376821878185, 0,
1.83194551960988828376821878185, 2.34337929206565910714894249260, 3.54616840774244918324171029601, 3.92114272928466676617818582957, 4.70074891919111595018573788134, 5.59011460426287360473884640673, 5.84635288065204829999376709079, 6.34428311741507214172984550860, 6.85017343928106467007064060318, 7.49744079817246587850200802404, 8.068071510872389098162573713865, 8.487995514415829589315400073807, 9.077417084015230940122177295816, 9.513325163360725882556033794995
