| L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s − 2·13-s − 2·17-s − 8·19-s − 4·21-s − 10·23-s − 6·27-s − 10·37-s + 4·39-s + 12·41-s + 6·43-s − 14·47-s + 2·49-s + 4·51-s − 2·53-s + 16·57-s − 8·59-s − 4·61-s + 4·63-s − 14·67-s + 20·69-s − 18·73-s − 16·79-s + 11·81-s − 10·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 0.872·21-s − 2.08·23-s − 1.15·27-s − 1.64·37-s + 0.640·39-s + 1.87·41-s + 0.914·43-s − 2.04·47-s + 2/7·49-s + 0.560·51-s − 0.274·53-s + 2.11·57-s − 1.04·59-s − 0.512·61-s + 0.503·63-s − 1.71·67-s + 2.40·69-s − 2.10·73-s − 1.80·79-s + 11/9·81-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.025637204908989211419191621093, −8.948862844657013768046769506814, −8.406837679272210433957835242484, −7.889536571156655130389814405102, −7.63282397342565243643138374584, −7.28854716051763515581253346644, −6.55940039953861065185824892891, −6.39077459417847177487225285081, −5.86575297264700713333873134176, −5.67345537595415015490197509243, −5.11131520097096109500924912143, −4.51785488697542070280998210141, −4.24887415762888992609219846950, −4.05225121294251739274196101591, −3.10534604220966593671916779248, −2.48054409441270760871203712167, −1.78081875858630517063994731620, −1.56730681119818120057759854084, 0, 0,
1.56730681119818120057759854084, 1.78081875858630517063994731620, 2.48054409441270760871203712167, 3.10534604220966593671916779248, 4.05225121294251739274196101591, 4.24887415762888992609219846950, 4.51785488697542070280998210141, 5.11131520097096109500924912143, 5.67345537595415015490197509243, 5.86575297264700713333873134176, 6.39077459417847177487225285081, 6.55940039953861065185824892891, 7.28854716051763515581253346644, 7.63282397342565243643138374584, 7.889536571156655130389814405102, 8.406837679272210433957835242484, 8.948862844657013768046769506814, 9.025637204908989211419191621093
