| L(s) = 1 | + 2·5-s + 4·7-s + 16-s − 2·17-s + 2·23-s + 25-s + 8·35-s − 2·43-s − 2·47-s + 8·49-s − 2·73-s + 2·80-s + 81-s + 4·83-s − 4·85-s + 4·112-s + 4·115-s − 8·119-s − 4·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2·5-s + 4·7-s + 16-s − 2·17-s + 2·23-s + 25-s + 8·35-s − 2·43-s − 2·47-s + 8·49-s − 2·73-s + 2·80-s + 81-s + 4·83-s − 4·85-s + 4·112-s + 4·115-s − 8·119-s − 4·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.575505740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575505740\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−6.96829873985163941699099799107, −6.39494738313924838814316052162, −6.35382912818920025163166578765, −6.15413903725315429259953379522, −5.95066718668048136819665674217, −5.73675234918728121096817945139, −5.39844900641203368948527819803, −5.13860173585919462425595207208, −5.12415546940069219469263240017, −4.83523607428323493651210226279, −4.72249620430373927388068272272, −4.61648928476780265969840442654, −4.58288894182124848905408330761, −3.84880483325986339020206104231, −3.80668435486843840396460578373, −3.40894672731197826436305030439, −3.30777871749459673314312383854, −2.69024534473089360619066240179, −2.41986216416759211326095713572, −2.32380069510543619947206085625, −1.95018773192803137022730556684, −1.90555224068989314331050168099, −1.36290705284242985829644560132, −1.28268344158568179228032519218, −1.23740987020273788573557037796,
1.23740987020273788573557037796, 1.28268344158568179228032519218, 1.36290705284242985829644560132, 1.90555224068989314331050168099, 1.95018773192803137022730556684, 2.32380069510543619947206085625, 2.41986216416759211326095713572, 2.69024534473089360619066240179, 3.30777871749459673314312383854, 3.40894672731197826436305030439, 3.80668435486843840396460578373, 3.84880483325986339020206104231, 4.58288894182124848905408330761, 4.61648928476780265969840442654, 4.72249620430373927388068272272, 4.83523607428323493651210226279, 5.12415546940069219469263240017, 5.13860173585919462425595207208, 5.39844900641203368948527819803, 5.73675234918728121096817945139, 5.95066718668048136819665674217, 6.15413903725315429259953379522, 6.35382912818920025163166578765, 6.39494738313924838814316052162, 6.96829873985163941699099799107
