| L(s) = 1 | + 2·7-s + 16-s − 4·19-s − 2·31-s + 4·37-s + 2·49-s + 2·67-s − 4·73-s − 2·97-s − 4·109-s + 2·112-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·7-s + 16-s − 4·19-s − 2·31-s + 4·37-s + 2·49-s + 2·67-s − 4·73-s − 2·97-s − 4·109-s + 2·112-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.064976214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064976214\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53266559401714879809711529465, −7.03584074913094836878907801725, −6.76262068206111734104709958169, −6.71277590789802506919211911853, −6.50016986121600570142966358012, −6.05684049972813905510002596884, −5.86413539278560194563176551196, −5.81480635589876521780908988675, −5.66557112231000737105543865605, −5.09483373265622154232903231747, −5.02500532180481058741405939992, −4.89047442243774104529053427569, −4.42567213703835788186160060856, −4.18745830780474049514533370521, −3.97560320878414841029199724586, −3.97286184087641713193403312142, −3.94853676393762341947859947883, −3.00308334381189184732855588110, −2.85903625350512672689144687492, −2.62726185557619559241107538370, −2.35737794471852370314795042979, −1.85339797520408494730768401042, −1.60720037013402787442658105614, −1.58959574356421557560574658293, −0.794864837539263430007271472500,
0.794864837539263430007271472500, 1.58959574356421557560574658293, 1.60720037013402787442658105614, 1.85339797520408494730768401042, 2.35737794471852370314795042979, 2.62726185557619559241107538370, 2.85903625350512672689144687492, 3.00308334381189184732855588110, 3.94853676393762341947859947883, 3.97286184087641713193403312142, 3.97560320878414841029199724586, 4.18745830780474049514533370521, 4.42567213703835788186160060856, 4.89047442243774104529053427569, 5.02500532180481058741405939992, 5.09483373265622154232903231747, 5.66557112231000737105543865605, 5.81480635589876521780908988675, 5.86413539278560194563176551196, 6.05684049972813905510002596884, 6.50016986121600570142966358012, 6.71277590789802506919211911853, 6.76262068206111734104709958169, 7.03584074913094836878907801725, 7.53266559401714879809711529465
