| L(s) = 1 | − 2·7-s − 9-s − 2·23-s + 2·25-s + 49-s + 2·63-s + 2·71-s + 8·79-s + 81-s + 4·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + 167-s + 169-s + 173-s − 4·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·7-s − 9-s − 2·23-s + 2·25-s + 49-s + 2·63-s + 2·71-s + 8·79-s + 81-s + 4·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + 167-s + 169-s + 173-s − 4·175-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9152937974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9152937974\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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$$\times$$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_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_1$ | \( ( 1 - T )^{8} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−6.54856986317789257674365671671, −6.02161818644375825341507672344, −6.02071818518047786439042659393, −5.93872402588197239084985667432, −5.87777850574482344496670255011, −5.25390642127094472153210000225, −5.03768080097442338164142911169, −5.02564635779060358465247931939, −5.01717781447296214306684575359, −4.62025652308720266794180091224, −4.24994251266490419694584502055, −4.04953968731682642527635413024, −3.70082965116310961415078413506, −3.66994068718345509775515915336, −3.41566279771880274089542340131, −3.27985271645970097095312946467, −3.05618063371112563605180064151, −2.65392681386733528411163941868, −2.61160197834719729589001989674, −2.12201618605801013601391304357, −2.01550396325368811208157216910, −1.92151531135703640762376054229, −1.17535486519201918524852363376, −0.72472539807507239207919086336, −0.57091494182057575968044174310,
0.57091494182057575968044174310, 0.72472539807507239207919086336, 1.17535486519201918524852363376, 1.92151531135703640762376054229, 2.01550396325368811208157216910, 2.12201618605801013601391304357, 2.61160197834719729589001989674, 2.65392681386733528411163941868, 3.05618063371112563605180064151, 3.27985271645970097095312946467, 3.41566279771880274089542340131, 3.66994068718345509775515915336, 3.70082965116310961415078413506, 4.04953968731682642527635413024, 4.24994251266490419694584502055, 4.62025652308720266794180091224, 5.01717781447296214306684575359, 5.02564635779060358465247931939, 5.03768080097442338164142911169, 5.25390642127094472153210000225, 5.87777850574482344496670255011, 5.93872402588197239084985667432, 6.02071818518047786439042659393, 6.02161818644375825341507672344, 6.54856986317789257674365671671
