| L(s) = 1 | + 9-s − 2·25-s + 6·41-s + 2·49-s − 4·73-s + 2·97-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2·225-s + ⋯ |
| L(s) = 1 | + 9-s − 2·25-s + 6·41-s + 2·49-s − 4·73-s + 2·97-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.542695509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542695509\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.56525597863044695490943265602, −6.25981445554587548154095423450, −6.04816706974671542623903465504, −6.01627085572829035552942210182, −5.93628350685361829433521148441, −5.57293075494166658867912107934, −5.42234042293614200634913079913, −5.07310533472400892319098449523, −5.06547542834992258641519991966, −4.41230288828844598445970233498, −4.33892791892797289599897993784, −4.30473027231193613378276981493, −4.20864469198996953266398000256, −3.89162785488803733323717631581, −3.55154673152441964930575617037, −3.52846938183535540873319029669, −2.94559019079676320023976896808, −2.74519341441249821020378530144, −2.54680630319399815907321155881, −2.24748419402585637127423440804, −2.22869031404860420791018203108, −1.58716519457094324597051653482, −1.39631768484973857788478946766, −1.09423754706444772878745002241, −0.63896316386965826928805405582,
0.63896316386965826928805405582, 1.09423754706444772878745002241, 1.39631768484973857788478946766, 1.58716519457094324597051653482, 2.22869031404860420791018203108, 2.24748419402585637127423440804, 2.54680630319399815907321155881, 2.74519341441249821020378530144, 2.94559019079676320023976896808, 3.52846938183535540873319029669, 3.55154673152441964930575617037, 3.89162785488803733323717631581, 4.20864469198996953266398000256, 4.30473027231193613378276981493, 4.33892791892797289599897993784, 4.41230288828844598445970233498, 5.06547542834992258641519991966, 5.07310533472400892319098449523, 5.42234042293614200634913079913, 5.57293075494166658867912107934, 5.93628350685361829433521148441, 6.01627085572829035552942210182, 6.04816706974671542623903465504, 6.25981445554587548154095423450, 6.56525597863044695490943265602
