| L(s) = 1 | − 4-s − 2·9-s + 2·13-s + 16-s − 2·19-s − 25-s − 2·31-s + 2·36-s − 2·37-s + 8·43-s − 4·49-s − 2·52-s + 2·61-s − 2·64-s + 2·67-s − 4·73-s + 2·76-s − 2·79-s + 3·81-s + 100-s + 2·109-s − 4·117-s − 121-s + 2·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 2·9-s + 2·13-s + 16-s − 2·19-s − 25-s − 2·31-s + 2·36-s − 2·37-s + 8·43-s − 4·49-s − 2·52-s + 2·61-s − 2·64-s + 2·67-s − 4·73-s + 2·76-s − 2·79-s + 3·81-s + 100-s + 2·109-s − 4·117-s − 121-s + 2·124-s + 127-s + 131-s + 137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 73^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 73^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1986670031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1986670031\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$ | \( ( 1 - T )^{8} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−9.115061166956220937802536284317, −9.031145638149597597615486186533, −8.784299478824379044515155798543, −8.454044486717555704184380014240, −8.346867242852283231094065261921, −8.145035225881083714886088642123, −7.60409008150833345560627439468, −7.49041836004249738006067837641, −7.29335706090320299408775243351, −6.77850921596049707941957554066, −6.21560617127001548551881573055, −6.12431870987516470758889059761, −5.97282719384216888726145132376, −5.78820432683777135754918197935, −5.43035803916290434209300285170, −5.18564009977154922100706772878, −4.62468068122705460477081580654, −4.32821436834920813882335256068, −3.93142438706968218785157178014, −3.72625818105839647164751979392, −3.58492594127133134992539910688, −2.83914978359970942486548318456, −2.63052913265292256292453604278, −2.04450464346036654471394195150, −1.38397290740702287558728443771,
1.38397290740702287558728443771, 2.04450464346036654471394195150, 2.63052913265292256292453604278, 2.83914978359970942486548318456, 3.58492594127133134992539910688, 3.72625818105839647164751979392, 3.93142438706968218785157178014, 4.32821436834920813882335256068, 4.62468068122705460477081580654, 5.18564009977154922100706772878, 5.43035803916290434209300285170, 5.78820432683777135754918197935, 5.97282719384216888726145132376, 6.12431870987516470758889059761, 6.21560617127001548551881573055, 6.77850921596049707941957554066, 7.29335706090320299408775243351, 7.49041836004249738006067837641, 7.60409008150833345560627439468, 8.145035225881083714886088642123, 8.346867242852283231094065261921, 8.454044486717555704184380014240, 8.784299478824379044515155798543, 9.031145638149597597615486186533, 9.115061166956220937802536284317
