| L(s) = 1 | + 2-s − 3-s − 2·5-s − 6-s − 7-s + 8-s − 2·9-s − 2·10-s − 2·13-s − 14-s + 2·15-s − 16-s − 2·18-s − 19-s + 21-s − 3·23-s − 24-s + 2·25-s − 2·26-s + 2·27-s − 6·29-s + 2·30-s − 15·31-s − 6·32-s + 2·35-s + 8·37-s − 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.471·18-s − 0.229·19-s + 0.218·21-s − 0.625·23-s − 0.204·24-s + 2/5·25-s − 0.392·26-s + 0.384·27-s − 1.11·29-s + 0.365·30-s − 2.69·31-s − 1.06·32-s + 0.338·35-s + 1.31·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28767 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28767 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 223 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 67 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 114 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 67 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 193 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 122 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 120 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−15.5270431844, −15.0657916090, −14.5100602034, −14.2718897911, −13.7873767175, −13.0105787173, −12.7788694301, −12.3982161832, −11.8583407480, −11.1551889256, −10.9947774527, −10.7314395548, −9.53853974742, −9.39261240246, −8.79406541295, −7.82293814832, −7.60742547241, −7.08648477458, −6.14567011474, −5.78611205243, −5.14943583926, −4.37778538316, −3.97577280222, −3.17821388227, −2.11192684504, 0,
2.11192684504, 3.17821388227, 3.97577280222, 4.37778538316, 5.14943583926, 5.78611205243, 6.14567011474, 7.08648477458, 7.60742547241, 7.82293814832, 8.79406541295, 9.39261240246, 9.53853974742, 10.7314395548, 10.9947774527, 11.1551889256, 11.8583407480, 12.3982161832, 12.7788694301, 13.0105787173, 13.7873767175, 14.2718897911, 14.5100602034, 15.0657916090, 15.5270431844
