| L(s) = 1 | − 2·4-s − 4·5-s − 9-s − 3·13-s + 4·16-s + 17-s + 8·20-s + 3·25-s − 10·29-s + 2·36-s + 11·37-s + 5·41-s + 4·45-s + 5·49-s + 6·52-s − 9·61-s − 8·64-s + 12·65-s − 2·68-s + 4·73-s − 16·80-s − 8·81-s − 4·85-s − 12·89-s + 13·97-s − 6·100-s + 8·101-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s − 1/3·9-s − 0.832·13-s + 16-s + 0.242·17-s + 1.78·20-s + 3/5·25-s − 1.85·29-s + 1/3·36-s + 1.80·37-s + 0.780·41-s + 0.596·45-s + 5/7·49-s + 0.832·52-s − 1.15·61-s − 64-s + 1.48·65-s − 0.242·68-s + 0.468·73-s − 1.78·80-s − 8/9·81-s − 0.433·85-s − 1.27·89-s + 1.31·97-s − 3/5·100-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 801424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 801424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 3853 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 11 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.916359405369457776771422113090, −7.72402820187893684182068562806, −7.37218545448228804549543467486, −6.91780679029660819275576962470, −5.96268646034225472105536489435, −5.83581464962084324009143458298, −5.19870678557801697388966911308, −4.61002578006593120831488946808, −4.21254444521202756811740390634, −3.92667756533440157861269238874, −3.34288787940312711540876419959, −2.82197554700405663189506578453, −1.92748487120488017080549424842, −0.74631737615818800366751742256, 0,
0.74631737615818800366751742256, 1.92748487120488017080549424842, 2.82197554700405663189506578453, 3.34288787940312711540876419959, 3.92667756533440157861269238874, 4.21254444521202756811740390634, 4.61002578006593120831488946808, 5.19870678557801697388966911308, 5.83581464962084324009143458298, 5.96268646034225472105536489435, 6.91780679029660819275576962470, 7.37218545448228804549543467486, 7.72402820187893684182068562806, 7.916359405369457776771422113090
