| L(s) = 1 | − 2·2-s + 2·7-s + 4·8-s − 4·14-s − 4·16-s − 19-s − 9·25-s + 2·29-s + 2·38-s − 4·41-s + 2·43-s − 7·49-s + 18·50-s + 10·53-s + 8·56-s − 4·58-s − 12·59-s − 2·61-s + 14·73-s + 8·82-s − 4·86-s + 10·89-s + 14·98-s − 20·106-s + 26·107-s − 8·112-s + 22·113-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.755·7-s + 1.41·8-s − 1.06·14-s − 16-s − 0.229·19-s − 9/5·25-s + 0.371·29-s + 0.324·38-s − 0.624·41-s + 0.304·43-s − 49-s + 2.54·50-s + 1.37·53-s + 1.06·56-s − 0.525·58-s − 1.56·59-s − 0.256·61-s + 1.63·73-s + 0.883·82-s − 0.431·86-s + 1.05·89-s + 1.41·98-s − 1.94·106-s + 2.51·107-s − 0.755·112-s + 2.06·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 81 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−8.317297547353168064398027241494, −7.966948918026342907375674265174, −7.49856236452372753309324500549, −7.26474360964016384572344893708, −6.36475506966271290067198780056, −6.10585756805282236654902083246, −5.41230934141795366211591142038, −4.79865079422132882343539337376, −4.58989237788303713873545179365, −3.85390392595623775470818960063, −3.38376554464490698480274175793, −2.33216445994765511920267044328, −1.78960940700603036241482449462, −1.00005550271132565748661288841, 0,
1.00005550271132565748661288841, 1.78960940700603036241482449462, 2.33216445994765511920267044328, 3.38376554464490698480274175793, 3.85390392595623775470818960063, 4.58989237788303713873545179365, 4.79865079422132882343539337376, 5.41230934141795366211591142038, 6.10585756805282236654902083246, 6.36475506966271290067198780056, 7.26474360964016384572344893708, 7.49856236452372753309324500549, 7.966948918026342907375674265174, 8.317297547353168064398027241494
