| L(s) = 1 | + 2-s + 4-s − 6·5-s + 8-s + 3·9-s − 6·10-s − 10·13-s + 16-s + 12·17-s + 3·18-s − 6·20-s + 17·25-s − 10·26-s + 12·29-s + 32-s + 12·34-s + 3·36-s − 20·37-s − 6·40-s + 4·41-s − 18·45-s − 5·49-s + 17·50-s − 10·52-s − 24·53-s + 12·58-s + 24·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 2.68·5-s + 0.353·8-s + 9-s − 1.89·10-s − 2.77·13-s + 1/4·16-s + 2.91·17-s + 0.707·18-s − 1.34·20-s + 17/5·25-s − 1.96·26-s + 2.22·29-s + 0.176·32-s + 2.05·34-s + 1/2·36-s − 3.28·37-s − 0.948·40-s + 0.624·41-s − 2.68·45-s − 5/7·49-s + 2.40·50-s − 1.38·52-s − 3.29·53-s + 1.57·58-s + 3.07·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
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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
−8.367530614402870047520128581958, −8.316300601517690619428978935480, −7.66138306465432851890178277032, −7.50874638191110861518479439117, −6.92116149525582118764831835825, −6.91576129806276774301284636347, −5.69924671275910195432130153357, −5.03332639383735802564822482656, −4.76932431908835403735443234199, −4.35064038074981651199670304457, −3.45774961440432575864801056504, −3.43543530646035424633948117891, −2.63887435377374729657972230954, −1.31619036555226546624498396933, 0,
1.31619036555226546624498396933, 2.63887435377374729657972230954, 3.43543530646035424633948117891, 3.45774961440432575864801056504, 4.35064038074981651199670304457, 4.76932431908835403735443234199, 5.03332639383735802564822482656, 5.69924671275910195432130153357, 6.91576129806276774301284636347, 6.92116149525582118764831835825, 7.50874638191110861518479439117, 7.66138306465432851890178277032, 8.316300601517690619428978935480, 8.367530614402870047520128581958
