| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·7-s − 8-s − 3·9-s − 2·10-s + 3·11-s − 12-s + 3·13-s − 3·14-s − 2·15-s − 16-s + 5·17-s − 3·18-s + 19-s + 2·20-s − 3·21-s + 3·22-s + 4·23-s − 24-s − 25-s + 3·26-s − 4·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 9-s − 0.632·10-s + 0.904·11-s − 0.288·12-s + 0.832·13-s − 0.801·14-s − 0.516·15-s − 1/4·16-s + 1.21·17-s − 0.707·18-s + 0.229·19-s + 0.447·20-s − 0.654·21-s + 0.639·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.588·26-s − 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 4013 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 36 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 4 T - 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 129 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 140 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T - 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T - 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−14.2387446701, −13.7331123357, −13.4344508174, −13.1719739110, −12.6181512177, −12.0738365006, −11.8210619525, −11.2002150933, −11.0015713733, −10.1652652696, −9.61983948740, −9.29582053760, −8.82587676910, −8.38596921385, −7.97434912286, −7.23510129072, −6.85176557018, −6.18541930020, −5.56536156527, −5.21946019589, −4.31975763962, −3.71827457469, −3.38378802645, −3.07240713677, −1.64509444758, 0,
1.64509444758, 3.07240713677, 3.38378802645, 3.71827457469, 4.31975763962, 5.21946019589, 5.56536156527, 6.18541930020, 6.85176557018, 7.23510129072, 7.97434912286, 8.38596921385, 8.82587676910, 9.29582053760, 9.61983948740, 10.1652652696, 11.0015713733, 11.2002150933, 11.8210619525, 12.0738365006, 12.6181512177, 13.1719739110, 13.4344508174, 13.7331123357, 14.2387446701
