| L(s) = 1 | − 3-s − 4-s + 3·7-s − 3·9-s − 3·11-s + 12-s − 3·13-s + 16-s + 3·17-s + 19-s − 3·21-s + 9·23-s − 2·25-s + 4·27-s − 3·28-s − 9·29-s + 3·33-s + 3·36-s − 12·37-s + 3·39-s + 9·41-s − 11·43-s + 3·44-s + 3·47-s − 48-s + 5·49-s − 3·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1.13·7-s − 9-s − 0.904·11-s + 0.288·12-s − 0.832·13-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.654·21-s + 1.87·23-s − 2/5·25-s + 0.769·27-s − 0.566·28-s − 1.67·29-s + 0.522·33-s + 1/2·36-s − 1.97·37-s + 0.480·39-s + 1.40·41-s − 1.67·43-s + 0.452·44-s + 0.437·47-s − 0.144·48-s + 5/7·49-s − 0.420·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104596 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104596 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| 331 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 32 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11 T + 107 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 92 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 198 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.3420126784, −13.8230447036, −13.3035110005, −12.9731318979, −12.4763838492, −11.8595769877, −11.6593201589, −11.2024781800, −10.7129777316, −10.3803771054, −9.83576204625, −9.13313061795, −8.86654225458, −8.32908354541, −7.77372899620, −7.38252831196, −6.94960924910, −5.97351617596, −5.44149856061, −5.21545131541, −4.86024955815, −3.97145722107, −3.17726275764, −2.52072535209, −1.44952846133, 0,
1.44952846133, 2.52072535209, 3.17726275764, 3.97145722107, 4.86024955815, 5.21545131541, 5.44149856061, 5.97351617596, 6.94960924910, 7.38252831196, 7.77372899620, 8.32908354541, 8.86654225458, 9.13313061795, 9.83576204625, 10.3803771054, 10.7129777316, 11.2024781800, 11.6593201589, 11.8595769877, 12.4763838492, 12.9731318979, 13.3035110005, 13.8230447036, 14.3420126784
