| L(s) = 1 | + 4-s + 2·7-s + 3·13-s + 16-s − 5·19-s − 4·25-s + 2·28-s − 7·31-s − 10·37-s − 12·43-s − 4·49-s + 3·52-s + 4·61-s + 64-s − 4·67-s + 14·73-s − 5·76-s − 10·79-s + 6·91-s − 4·100-s − 16·103-s + 19·109-s + 2·112-s − 8·121-s − 7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 0.832·13-s + 1/4·16-s − 1.14·19-s − 4/5·25-s + 0.377·28-s − 1.25·31-s − 1.64·37-s − 1.82·43-s − 4/7·49-s + 0.416·52-s + 0.512·61-s + 1/8·64-s − 0.488·67-s + 1.63·73-s − 0.573·76-s − 1.12·79-s + 0.628·91-s − 2/5·100-s − 1.57·103-s + 1.81·109-s + 0.188·112-s − 0.727·121-s − 0.628·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 397 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 32 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.972940309086356299790602818487, −7.61592734302030268389409486485, −6.98069358270179228607489413874, −6.65895476160480322150220904586, −6.27279910046810308876547843013, −5.68268240949715230044898189687, −5.26233575643732078832303789631, −4.85192395731416778565354446148, −4.15382609367416340580520692377, −3.69158333464770211879161427825, −3.29111780589789193454818766884, −2.41095286094252482860781666652, −1.81880823420482889757961013654, −1.43579326087821225968750092192, 0,
1.43579326087821225968750092192, 1.81880823420482889757961013654, 2.41095286094252482860781666652, 3.29111780589789193454818766884, 3.69158333464770211879161427825, 4.15382609367416340580520692377, 4.85192395731416778565354446148, 5.26233575643732078832303789631, 5.68268240949715230044898189687, 6.27279910046810308876547843013, 6.65895476160480322150220904586, 6.98069358270179228607489413874, 7.61592734302030268389409486485, 7.972940309086356299790602818487
