| L(s) = 1 | + 4-s + 2·5-s + 2·17-s + 2·20-s + 2·25-s + 4·29-s + 2·37-s + 4·41-s + 2·61-s − 64-s + 2·68-s − 2·73-s + 81-s + 4·85-s − 4·97-s + 2·100-s + 2·109-s − 4·113-s + 4·116-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 2·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s + 2·5-s + 2·17-s + 2·20-s + 2·25-s + 4·29-s + 2·37-s + 4·41-s + 2·61-s − 64-s + 2·68-s − 2·73-s + 81-s + 4·85-s − 4·97-s + 2·100-s + 2·109-s − 4·113-s + 4·116-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 2·148-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.244973700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.244973700\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.30005263831985852621532516100, −5.99452786644814829947380513637, −5.98688946444183971461784402312, −5.69567098547523785107523463382, −5.67743266210846920284039915086, −5.47919858997078262523898582176, −4.98403223714129092548207534984, −4.97161127033563395014800418316, −4.76371918689662280376998643810, −4.59377084417835667756505715140, −4.30681260624239830956482994031, −3.91879664829420711408518867411, −3.80382516343326778635040671762, −3.76177156672170888550589658377, −3.19403757196379110927894446113, −2.85335583804944230706555168065, −2.77339437860753291328789928088, −2.55805414342024236826004287282, −2.45050393713236533417445482065, −2.44091938312418767833749439081, −2.02057573513368101683399016705, −1.31662503215483504219125268092, −1.23450432191270852472600508450, −1.14770475034360088537504572959, −1.02106786007784933774402506917,
1.02106786007784933774402506917, 1.14770475034360088537504572959, 1.23450432191270852472600508450, 1.31662503215483504219125268092, 2.02057573513368101683399016705, 2.44091938312418767833749439081, 2.45050393713236533417445482065, 2.55805414342024236826004287282, 2.77339437860753291328789928088, 2.85335583804944230706555168065, 3.19403757196379110927894446113, 3.76177156672170888550589658377, 3.80382516343326778635040671762, 3.91879664829420711408518867411, 4.30681260624239830956482994031, 4.59377084417835667756505715140, 4.76371918689662280376998643810, 4.97161127033563395014800418316, 4.98403223714129092548207534984, 5.47919858997078262523898582176, 5.67743266210846920284039915086, 5.69567098547523785107523463382, 5.98688946444183971461784402312, 5.99452786644814829947380513637, 6.30005263831985852621532516100
