| L(s) = 1 | − 2·9-s + 16-s − 2·19-s − 2·31-s + 2·37-s − 6·43-s − 4·67-s − 4·73-s + 3·81-s + 2·97-s − 2·103-s − 2·109-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 4·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·9-s + 16-s − 2·19-s − 2·31-s + 2·37-s − 6·43-s − 4·67-s − 4·73-s + 3·81-s + 2·97-s − 2·103-s − 2·109-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 4·171-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08818080917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08818080917\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_1$$\times$$C_2^2$ | \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \) |
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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
−6.88005798180805097150728296988, −6.32002768437934809944348336284, −6.25722816806727583209750234206, −6.24475156984861357190844318170, −5.81973955234642667384501560513, −5.73588400531937188413042981378, −5.63125842014480916359814700242, −5.34713816254136412432785363583, −4.91462636011125901705060056737, −4.89760807258971427383140061147, −4.75978207064520881903208103721, −4.26581400444214462470810069566, −4.12102083222920246905467550340, −4.07576047968334170661709160599, −3.40677505305503168240067982979, −3.38228421077878209110660240298, −3.21853185773326238921527221135, −2.89103132590686388915593419621, −2.87668400731527366152353534366, −2.34683766658214896477436812572, −1.94568860600968067248310252991, −1.94045938752911281675430471050, −1.36552406150698187879630309829, −1.34123154517580344023955600913, −0.14485906872427889621092487023,
0.14485906872427889621092487023, 1.34123154517580344023955600913, 1.36552406150698187879630309829, 1.94045938752911281675430471050, 1.94568860600968067248310252991, 2.34683766658214896477436812572, 2.87668400731527366152353534366, 2.89103132590686388915593419621, 3.21853185773326238921527221135, 3.38228421077878209110660240298, 3.40677505305503168240067982979, 4.07576047968334170661709160599, 4.12102083222920246905467550340, 4.26581400444214462470810069566, 4.75978207064520881903208103721, 4.89760807258971427383140061147, 4.91462636011125901705060056737, 5.34713816254136412432785363583, 5.63125842014480916359814700242, 5.73588400531937188413042981378, 5.81973955234642667384501560513, 6.24475156984861357190844318170, 6.25722816806727583209750234206, 6.32002768437934809944348336284, 6.88005798180805097150728296988
