| L(s) = 1 | + 2-s + 4-s − 7-s + 2·8-s + 11-s − 14-s + 2·16-s + 22-s − 2·23-s − 25-s − 28-s − 2·29-s + 2·32-s − 2·37-s + 43-s + 44-s − 2·46-s − 50-s − 2·53-s − 2·56-s − 2·58-s + 3·64-s + 67-s − 2·71-s − 2·74-s − 77-s + 79-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 7-s + 2·8-s + 11-s − 14-s + 2·16-s + 22-s − 2·23-s − 25-s − 28-s − 2·29-s + 2·32-s − 2·37-s + 43-s + 44-s − 2·46-s − 50-s − 2·53-s − 2·56-s − 2·58-s + 3·64-s + 67-s − 2·71-s − 2·74-s − 77-s + 79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.469821136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469821136\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−11.03894343978437635354470649680, −11.01213510776796008077466787828, −10.26492597290797364311265623501, −9.965614092200162167904342640292, −9.578563332375894758161409830910, −9.126884492599423864605666810890, −8.425168318342960032448786383955, −7.922766317425265524038110269641, −7.39511579303658622609112329814, −7.21691880131830335540995874591, −6.47272779143865155221813273215, −6.17336019709093762902976537575, −5.74549199459339738671089215640, −5.17074570175580096443816158640, −4.49668122754269432802427301427, −3.86900875947489756213189227700, −3.76844469612622508811687578974, −3.07743310728371945458667310491, −1.89819266723662397843145634222, −1.80881972249632096017421423116,
1.80881972249632096017421423116, 1.89819266723662397843145634222, 3.07743310728371945458667310491, 3.76844469612622508811687578974, 3.86900875947489756213189227700, 4.49668122754269432802427301427, 5.17074570175580096443816158640, 5.74549199459339738671089215640, 6.17336019709093762902976537575, 6.47272779143865155221813273215, 7.21691880131830335540995874591, 7.39511579303658622609112329814, 7.922766317425265524038110269641, 8.425168318342960032448786383955, 9.126884492599423864605666810890, 9.578563332375894758161409830910, 9.965614092200162167904342640292, 10.26492597290797364311265623501, 11.01213510776796008077466787828, 11.03894343978437635354470649680
