| L(s) = 1 | + 2·3-s − 4-s − 5-s + 2·7-s + 2·8-s − 9-s − 2·12-s − 3·13-s − 2·15-s + 16-s + 2·17-s + 2·19-s + 20-s + 4·21-s − 12·23-s + 4·24-s − 3·25-s − 6·27-s − 2·28-s − 6·29-s − 4·32-s − 2·35-s + 36-s + 4·37-s − 6·39-s − 2·40-s + 7·41-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.707·8-s − 1/3·9-s − 0.577·12-s − 0.832·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.458·19-s + 0.223·20-s + 0.872·21-s − 2.50·23-s + 0.816·24-s − 3/5·25-s − 1.15·27-s − 0.377·28-s − 1.11·29-s − 0.707·32-s − 0.338·35-s + 1/6·36-s + 0.657·37-s − 0.960·39-s − 0.316·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531122 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531122 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 265561 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 380 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 71 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 82 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 77 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−12.8799256890, −12.4062381063, −11.9736746603, −11.6431706096, −11.2696546397, −10.7649198725, −10.3305311771, −9.75641808591, −9.50241140933, −9.20569678756, −8.45263044463, −8.26304486745, −7.81670232000, −7.58723779472, −7.33455629161, −6.38584537044, −5.73665191728, −5.54670237934, −4.82222940156, −4.28059220259, −3.88485412034, −3.46560812936, −2.54407488837, −2.23348061678, −1.40111456300, 0,
1.40111456300, 2.23348061678, 2.54407488837, 3.46560812936, 3.88485412034, 4.28059220259, 4.82222940156, 5.54670237934, 5.73665191728, 6.38584537044, 7.33455629161, 7.58723779472, 7.81670232000, 8.26304486745, 8.45263044463, 9.20569678756, 9.50241140933, 9.75641808591, 10.3305311771, 10.7649198725, 11.2696546397, 11.6431706096, 11.9736746603, 12.4062381063, 12.8799256890
