L(s) = 1 | + 3-s − 3·11-s − 2·19-s − 25-s − 27-s − 3·33-s + 3·41-s + 43-s + 49-s − 2·57-s + 3·59-s − 67-s + 2·73-s − 75-s − 81-s − 97-s + 5·121-s + 3·123-s + 127-s + 129-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | + 3-s − 3·11-s − 2·19-s − 25-s − 27-s − 3·33-s + 3·41-s + 43-s + 49-s − 2·57-s + 3·59-s − 67-s + 2·73-s − 75-s − 81-s − 97-s + 5·121-s + 3·123-s + 127-s + 129-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.035553082\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035553082\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.605767375930429728587670968479, −8.893744372641467252515801184243, −8.533644950631859568369830404256, −8.159320852021581715743440679313, −7.962148979373950399455754520165, −7.54828765323901464157354074993, −7.39572366331204352982975603054, −6.73502732585215328951843314824, −6.19637696979288174868091378008, −5.77087602296867134112040487781, −5.46929561747886872124164907651, −5.15227247524318535371991892579, −4.46461608813179270430920884300, −4.00498221469331237917395785908, −3.84415029382883093170813995687, −2.82503222881795133443596256572, −2.72899768943759094528764426145, −2.24050854121712394295276920377, −2.04920823029025466820722158425, −0.60806986647506836823907979026,
0.60806986647506836823907979026, 2.04920823029025466820722158425, 2.24050854121712394295276920377, 2.72899768943759094528764426145, 2.82503222881795133443596256572, 3.84415029382883093170813995687, 4.00498221469331237917395785908, 4.46461608813179270430920884300, 5.15227247524318535371991892579, 5.46929561747886872124164907651, 5.77087602296867134112040487781, 6.19637696979288174868091378008, 6.73502732585215328951843314824, 7.39572366331204352982975603054, 7.54828765323901464157354074993, 7.962148979373950399455754520165, 8.159320852021581715743440679313, 8.533644950631859568369830404256, 8.893744372641467252515801184243, 9.605767375930429728587670968479