| L(s) = 1 | − 2-s + 2·3-s − 2·4-s + 2·5-s − 2·6-s + 3·8-s − 3·9-s − 2·10-s − 4·12-s + 4·13-s + 4·15-s + 16-s + 3·18-s + 10·19-s − 4·20-s − 2·23-s + 6·24-s − 2·25-s − 4·26-s − 14·27-s + 6·29-s − 4·30-s + 18·31-s − 2·32-s + 6·36-s + 2·37-s − 10·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s + 0.894·5-s − 0.816·6-s + 1.06·8-s − 9-s − 0.632·10-s − 1.15·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s + 0.707·18-s + 2.29·19-s − 0.894·20-s − 0.417·23-s + 1.22·24-s − 2/5·25-s − 0.784·26-s − 2.69·27-s + 1.11·29-s − 0.730·30-s + 3.23·31-s − 0.353·32-s + 36-s + 0.328·37-s − 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1270129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1270129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990876527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990876527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 201 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.860379609929391644596808189317, −9.600605435854286737054013622807, −9.104390229178503025244814392006, −8.745639440852546702786070108862, −8.529760048167120976518761716965, −8.138474069919238163754920597156, −7.991131461537774865548882492784, −7.30648981782828696847078997561, −6.79856321375055198010680089203, −6.08175246265480076408242380055, −5.69177561923566685682885455384, −5.63520962271459585262950663416, −4.65435797979836386906048392244, −4.54363451617843676591953035776, −3.54304263870856834504283365428, −3.39132672155244407036368218894, −2.66432946802710409905076536200, −2.35284789215438961085117919597, −1.27503629514242313524015434312, −0.75480863510713876683498937908,
0.75480863510713876683498937908, 1.27503629514242313524015434312, 2.35284789215438961085117919597, 2.66432946802710409905076536200, 3.39132672155244407036368218894, 3.54304263870856834504283365428, 4.54363451617843676591953035776, 4.65435797979836386906048392244, 5.63520962271459585262950663416, 5.69177561923566685682885455384, 6.08175246265480076408242380055, 6.79856321375055198010680089203, 7.30648981782828696847078997561, 7.991131461537774865548882492784, 8.138474069919238163754920597156, 8.529760048167120976518761716965, 8.745639440852546702786070108862, 9.104390229178503025244814392006, 9.600605435854286737054013622807, 9.860379609929391644596808189317
