| L(s) = 1 | − 2-s + 4-s + 4·5-s + 2·7-s + 2·8-s − 4·10-s − 4·11-s − 2·14-s + 8·17-s + 4·19-s + 4·20-s + 4·22-s − 6·23-s + 10·25-s + 2·28-s + 2·29-s − 16·31-s + 4·32-s − 8·34-s + 8·35-s − 4·38-s + 8·40-s + 6·41-s + 6·43-s − 4·44-s + 6·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.755·7-s + 0.707·8-s − 1.26·10-s − 1.20·11-s − 0.534·14-s + 1.94·17-s + 0.917·19-s + 0.894·20-s + 0.852·22-s − 1.25·23-s + 2·25-s + 0.377·28-s + 0.371·29-s − 2.87·31-s + 0.707·32-s − 1.37·34-s + 1.35·35-s − 0.648·38-s + 1.26·40-s + 0.937·41-s + 0.914·43-s − 0.603·44-s + 0.884·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.590334981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.590334981\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 36 T^{3} - 128 T^{4} - 36 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 27 T^{2} - 24 T^{3} - 8 T^{4} - 24 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 13 T^{2} + 36 T^{3} + 176 T^{4} + 36 p T^{5} - 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 102 T^{3} - 908 T^{4} + 102 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 61 T^{2} + 2352 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 7 T^{2} + 6 p T^{3} - 36 p T^{4} + 6 p^{2} T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{2} - 3480 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 83 T^{2} - 972 T^{3} - 11544 T^{4} - 972 p T^{5} + 83 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2 T - 123 T^{2} + 102 T^{3} + 7852 T^{4} + 102 p T^{5} - 123 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24 T + 251 T^{2} - 3144 T^{3} + 40344 T^{4} - 3144 p T^{5} + 251 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.75921891865245988950491983430, −7.74095564126832266571094937208, −7.18248499983267701018513766325, −7.07878304512902982558008474063, −6.94035521265730718962614189013, −6.65219663898547700268366944517, −6.04239136989103537315520057443, −5.92618970641545711914155846601, −5.77434285912283199724715768955, −5.50098449856334993908599171837, −5.46786814581091082008414050134, −5.22083801248646958759740155185, −4.85654305130053525472655843335, −4.37578571209581520864890986887, −4.37498755968060647550406637600, −3.93897582977018517080391798143, −3.50539217898122236647920480363, −3.03076952136180675816034940387, −3.01556184242093139969566683202, −2.61519630802696832619676073954, −2.13491309999883799829587654311, −1.77661219824126777575458276923, −1.57469186591340619258495620963, −1.30514420746995602798904358704, −0.64903909413762641751616505569,
0.64903909413762641751616505569, 1.30514420746995602798904358704, 1.57469186591340619258495620963, 1.77661219824126777575458276923, 2.13491309999883799829587654311, 2.61519630802696832619676073954, 3.01556184242093139969566683202, 3.03076952136180675816034940387, 3.50539217898122236647920480363, 3.93897582977018517080391798143, 4.37498755968060647550406637600, 4.37578571209581520864890986887, 4.85654305130053525472655843335, 5.22083801248646958759740155185, 5.46786814581091082008414050134, 5.50098449856334993908599171837, 5.77434285912283199724715768955, 5.92618970641545711914155846601, 6.04239136989103537315520057443, 6.65219663898547700268366944517, 6.94035521265730718962614189013, 7.07878304512902982558008474063, 7.18248499983267701018513766325, 7.74095564126832266571094937208, 7.75921891865245988950491983430
