| L(s) = 1 | + 2-s − 5·7-s + 8-s + 2·11-s − 10·13-s − 5·14-s − 16-s + 3·17-s − 2·19-s + 2·22-s − 11·23-s − 10·26-s + 9·29-s + 6·31-s − 6·32-s + 3·34-s − 12·37-s − 2·38-s + 4·41-s − 11·46-s − 6·47-s + 8·49-s + 53-s − 5·56-s + 9·58-s + 14·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.88·7-s + 0.353·8-s + 0.603·11-s − 2.77·13-s − 1.33·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.426·22-s − 2.29·23-s − 1.96·26-s + 1.67·29-s + 1.07·31-s − 1.06·32-s + 0.514·34-s − 1.97·37-s − 0.324·38-s + 0.624·41-s − 1.62·46-s − 0.875·47-s + 8/7·49-s + 0.137·53-s − 0.668·56-s + 1.18·58-s + 1.82·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 11 T + 73 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 115 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 159 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 187 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 373 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
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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
−8.591268366849093939615662148969, −8.415399862659603199416534755798, −7.986651192879807112285336444773, −7.27442983818976872671281423815, −7.25282030245603725073723069555, −6.73660496108074342696780738501, −6.38967263571840803143135749724, −6.19364040526483233730905915247, −5.37353016843940857903175698802, −5.31495082266845010545941783590, −4.58610882863565380133372931648, −4.49289121151963948870230729306, −3.75120802667488387193287067867, −3.65481076245711625675108826063, −2.91199382599288205957143941015, −2.47159096994729384411981162893, −2.21239638480208954961775680034, −1.28350279549380974401560284608, 0, 0,
1.28350279549380974401560284608, 2.21239638480208954961775680034, 2.47159096994729384411981162893, 2.91199382599288205957143941015, 3.65481076245711625675108826063, 3.75120802667488387193287067867, 4.49289121151963948870230729306, 4.58610882863565380133372931648, 5.31495082266845010545941783590, 5.37353016843940857903175698802, 6.19364040526483233730905915247, 6.38967263571840803143135749724, 6.73660496108074342696780738501, 7.25282030245603725073723069555, 7.27442983818976872671281423815, 7.986651192879807112285336444773, 8.415399862659603199416534755798, 8.591268366849093939615662148969
