| L(s) = 1 | + 2·2-s + 4-s − 2·8-s + 9-s − 4·16-s + 2·18-s − 4·23-s + 2·25-s − 2·32-s + 36-s − 8·46-s + 4·50-s + 3·64-s + 4·71-s − 2·72-s − 2·79-s − 4·92-s + 2·100-s − 2·113-s + 4·121-s + 127-s + 6·128-s + 131-s + 137-s + 139-s + 8·142-s − 4·144-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·8-s + 9-s − 4·16-s + 2·18-s − 4·23-s + 2·25-s − 2·32-s + 36-s − 8·46-s + 4·50-s + 3·64-s + 4·71-s − 2·72-s − 2·79-s − 4·92-s + 2·100-s − 2·113-s + 4·121-s + 127-s + 6·128-s + 131-s + 137-s + 139-s + 8·142-s − 4·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.887151264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.887151264\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−6.19523619932347851100429118368, −6.16337252331402888312046417073, −5.78199537281687662420233266465, −5.48816836574634368656535341348, −5.41886377357936195570998954984, −5.31918430281186626348800863591, −4.97814303889992368844988227261, −4.72533628252201551206491413162, −4.69738356936849020945953159931, −4.37909799265765769939258035681, −4.27100168184256196346688372650, −3.94979833021120617646471566025, −3.93420411185966584091009625722, −3.76637709384752569439902019718, −3.39510869976281655838227019726, −3.34264099669309637269865158455, −2.96386239084847630046122890208, −2.77317902782277587216845953945, −2.38682055650684543456572838803, −2.36060904184755265515560302030, −1.99725985665076023628412789238, −1.66370152428346974851814850829, −1.49284217269098366485974919895, −0.828639350533441648242716140987, −0.52295272085902699791821479950,
0.52295272085902699791821479950, 0.828639350533441648242716140987, 1.49284217269098366485974919895, 1.66370152428346974851814850829, 1.99725985665076023628412789238, 2.36060904184755265515560302030, 2.38682055650684543456572838803, 2.77317902782277587216845953945, 2.96386239084847630046122890208, 3.34264099669309637269865158455, 3.39510869976281655838227019726, 3.76637709384752569439902019718, 3.93420411185966584091009625722, 3.94979833021120617646471566025, 4.27100168184256196346688372650, 4.37909799265765769939258035681, 4.69738356936849020945953159931, 4.72533628252201551206491413162, 4.97814303889992368844988227261, 5.31918430281186626348800863591, 5.41886377357936195570998954984, 5.48816836574634368656535341348, 5.78199537281687662420233266465, 6.16337252331402888312046417073, 6.19523619932347851100429118368
