| L(s) = 1 | + 2-s − 2·5-s + 7-s − 8-s − 2·10-s + 4·11-s + 2·13-s + 14-s − 16-s + 4·22-s + 8·23-s + 3·25-s + 2·26-s − 29-s − 10·31-s − 2·35-s − 8·37-s + 2·40-s + 5·41-s + 43-s + 8·46-s − 13·47-s − 6·49-s + 3·50-s − 10·53-s − 8·55-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.852·22-s + 1.66·23-s + 3/5·25-s + 0.392·26-s − 0.185·29-s − 1.79·31-s − 0.338·35-s − 1.31·37-s + 0.316·40-s + 0.780·41-s + 0.152·43-s + 1.17·46-s − 1.89·47-s − 6/7·49-s + 0.424·50-s − 1.37·53-s − 1.07·55-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.430963932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.430963932\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 5 T - 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.186686693228526530007088105559, −9.105858355216914750637008423527, −8.722393517566310499101450184728, −8.065875346397218942162216694630, −8.005204749049220619957158850133, −7.39659280617023228902409937997, −6.95280438904286353898789506918, −6.55579551464440109994927815109, −6.46838490975301797217630433031, −5.53595592890952591894685615986, −5.43567660157689075716291411992, −4.87361233079471273551122740156, −4.50841482253130390221771947794, −3.88511839869504617121733420478, −3.80200228736646301577569240093, −3.07257159113219555058402957706, −2.97101924332474091953542326962, −1.68737987977483384412346812323, −1.57430093166884946750698282387, −0.52320711565888108087067980422,
0.52320711565888108087067980422, 1.57430093166884946750698282387, 1.68737987977483384412346812323, 2.97101924332474091953542326962, 3.07257159113219555058402957706, 3.80200228736646301577569240093, 3.88511839869504617121733420478, 4.50841482253130390221771947794, 4.87361233079471273551122740156, 5.43567660157689075716291411992, 5.53595592890952591894685615986, 6.46838490975301797217630433031, 6.55579551464440109994927815109, 6.95280438904286353898789506918, 7.39659280617023228902409937997, 8.005204749049220619957158850133, 8.065875346397218942162216694630, 8.722393517566310499101450184728, 9.105858355216914750637008423527, 9.186686693228526530007088105559
