L(s) = 1 | − 2·3-s − 5-s + 3·9-s − 4·13-s + 2·15-s − 6·17-s − 4·19-s − 6·23-s − 10·27-s + 12·29-s − 4·31-s − 2·37-s + 8·39-s − 12·41-s − 20·43-s − 3·45-s − 6·47-s + 12·51-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s + 4·65-s − 2·67-s + 12·69-s − 24·71-s + 2·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 9-s − 1.10·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 1.92·27-s + 2.22·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s − 1.87·41-s − 3.04·43-s − 0.447·45-s − 0.875·47-s + 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 0.244·67-s + 1.44·69-s − 2.84·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.03965015248128628912106807092, −9.609312979669962421036077568725, −8.765618259406640218291553476730, −8.469029290848132533946342079075, −8.346358474554124649413682515239, −7.55063861849093784405400132968, −7.00697775460501886819900456835, −6.96309481843633676754822466877, −6.22388277960908917403967252649, −6.18254081660402112783335453292, −5.27782986439097672900272675997, −5.01832465489147199907528245066, −4.56146916215330830942690266943, −4.12663907617630997213205759291, −3.59689222064108361214940628665, −2.81707654877198533725577674054, −2.05404859426744015397564861568, −1.59507968108844440572611551061, 0, 0,
1.59507968108844440572611551061, 2.05404859426744015397564861568, 2.81707654877198533725577674054, 3.59689222064108361214940628665, 4.12663907617630997213205759291, 4.56146916215330830942690266943, 5.01832465489147199907528245066, 5.27782986439097672900272675997, 6.18254081660402112783335453292, 6.22388277960908917403967252649, 6.96309481843633676754822466877, 7.00697775460501886819900456835, 7.55063861849093784405400132968, 8.346358474554124649413682515239, 8.469029290848132533946342079075, 8.765618259406640218291553476730, 9.609312979669962421036077568725, 10.03965015248128628912106807092