| L(s) = 1 | + 2·3-s − 3·4-s + 3·9-s − 6·12-s − 2·13-s + 5·16-s − 12·17-s − 6·25-s + 4·27-s − 4·29-s − 9·36-s − 4·39-s − 8·43-s + 10·48-s + 49-s − 24·51-s + 6·52-s + 12·53-s − 4·61-s − 3·64-s + 36·68-s − 12·75-s − 32·79-s + 5·81-s − 8·87-s + 18·100-s + 28·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 9-s − 1.73·12-s − 0.554·13-s + 5/4·16-s − 2.91·17-s − 6/5·25-s + 0.769·27-s − 0.742·29-s − 3/2·36-s − 0.640·39-s − 1.21·43-s + 1.44·48-s + 1/7·49-s − 3.36·51-s + 0.832·52-s + 1.64·53-s − 0.512·61-s − 3/8·64-s + 4.36·68-s − 1.38·75-s − 3.60·79-s + 5/9·81-s − 0.857·87-s + 9/5·100-s + 2.78·101-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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.542092147710694875643429105654, −8.829118090640207176685575408265, −8.672365196683144779961098572501, −8.407377139641223631410053626523, −7.48728380165555641417746344077, −7.25047783802838427330028450541, −6.50644747922743051012510731425, −5.81934189952643673303329908812, −5.00585213491191130205663070545, −4.48166630149180553918760034444, −4.13559084050773741974089362056, −3.51954643368349943520882598505, −2.53133608948261501639635653970, −1.88453572273281662045561783260, 0,
1.88453572273281662045561783260, 2.53133608948261501639635653970, 3.51954643368349943520882598505, 4.13559084050773741974089362056, 4.48166630149180553918760034444, 5.00585213491191130205663070545, 5.81934189952643673303329908812, 6.50644747922743051012510731425, 7.25047783802838427330028450541, 7.48728380165555641417746344077, 8.407377139641223631410053626523, 8.672365196683144779961098572501, 8.829118090640207176685575408265, 9.542092147710694875643429105654
