| L(s) = 1 | − 4-s + 3·7-s − 9-s + 16-s + 2·19-s + 23-s + 8·25-s − 3·28-s − 5·29-s + 36-s − 12·37-s + 10·41-s − 3·43-s − 4·47-s − 10·53-s − 3·59-s + 61-s − 3·63-s − 64-s − 2·76-s − 8·81-s − 2·83-s − 92-s − 23·97-s − 8·100-s − 21·101-s − 10·107-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.13·7-s − 1/3·9-s + 1/4·16-s + 0.458·19-s + 0.208·23-s + 8/5·25-s − 0.566·28-s − 0.928·29-s + 1/6·36-s − 1.97·37-s + 1.56·41-s − 0.457·43-s − 0.583·47-s − 1.37·53-s − 0.390·59-s + 0.128·61-s − 0.377·63-s − 1/8·64-s − 0.229·76-s − 8/9·81-s − 0.219·83-s − 0.104·92-s − 2.33·97-s − 4/5·100-s − 2.08·101-s − 0.966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1086652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1086652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 197 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.945997269573972208066346538718, −7.51918358020720898222136824550, −6.94552985403136397263134258930, −6.69666626617898731223547132994, −5.99576649115442169574688285638, −5.45487654721390936634831387046, −5.13093921864564077466531776454, −4.82821191658844945395631734500, −4.20911127538073657102574959220, −3.74676973692276842749667867399, −3.04520352638174430175056099719, −2.64431354894229359502802333880, −1.64068822062398056523197464723, −1.27904392882035021887078309359, 0,
1.27904392882035021887078309359, 1.64068822062398056523197464723, 2.64431354894229359502802333880, 3.04520352638174430175056099719, 3.74676973692276842749667867399, 4.20911127538073657102574959220, 4.82821191658844945395631734500, 5.13093921864564077466531776454, 5.45487654721390936634831387046, 5.99576649115442169574688285638, 6.69666626617898731223547132994, 6.94552985403136397263134258930, 7.51918358020720898222136824550, 7.945997269573972208066346538718
