| L(s) = 1 | + 2·3-s + 4-s − 6·5-s − 3·9-s + 6·11-s + 2·12-s − 12·15-s + 16-s − 6·20-s + 17·25-s − 14·27-s − 8·31-s + 12·33-s − 3·36-s − 14·37-s + 6·44-s + 18·45-s + 6·47-s + 2·48-s − 13·49-s − 36·55-s − 12·59-s − 12·60-s + 64-s + 28·67-s − 6·71-s + 34·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 2.68·5-s − 9-s + 1.80·11-s + 0.577·12-s − 3.09·15-s + 1/4·16-s − 1.34·20-s + 17/5·25-s − 2.69·27-s − 1.43·31-s + 2.08·33-s − 1/2·36-s − 2.30·37-s + 0.904·44-s + 2.68·45-s + 0.875·47-s + 0.288·48-s − 1.85·49-s − 4.85·55-s − 1.56·59-s − 1.54·60-s + 1/8·64-s + 3.42·67-s − 0.712·71-s + 3.92·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81796 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81796 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.252086119982469784390910809299, −8.817891777012169144954253302070, −8.289125051595985253760732717123, −8.158308298923332532258561346975, −7.61357841654986180404055851494, −6.99995607550208376322691762678, −6.76423132327857768444846951818, −5.85883048660862822422904238763, −5.16078862035801732240851587336, −4.20840140522021551558258294497, −3.64028761626013442697591583469, −3.58506615738380785640972516083, −2.89028538035854279081544483316, −1.75251405579569256697321594310, 0,
1.75251405579569256697321594310, 2.89028538035854279081544483316, 3.58506615738380785640972516083, 3.64028761626013442697591583469, 4.20840140522021551558258294497, 5.16078862035801732240851587336, 5.85883048660862822422904238763, 6.76423132327857768444846951818, 6.99995607550208376322691762678, 7.61357841654986180404055851494, 8.158308298923332532258561346975, 8.289125051595985253760732717123, 8.817891777012169144954253302070, 9.252086119982469784390910809299
