| L(s) = 1 | − 2·2-s − 2·3-s − 2·5-s + 4·6-s − 2·7-s + 4·8-s + 9-s + 4·10-s − 2·11-s − 6·13-s + 4·14-s + 4·15-s − 4·16-s + 6·17-s − 2·18-s + 2·19-s + 4·21-s + 4·22-s + 8·23-s − 8·24-s − 6·25-s + 12·26-s − 2·27-s − 8·30-s − 8·31-s + 4·33-s − 12·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 1.66·13-s + 1.06·14-s + 1.03·15-s − 16-s + 1.45·17-s − 0.471·18-s + 0.458·19-s + 0.872·21-s + 0.852·22-s + 1.66·23-s − 1.63·24-s − 6/5·25-s + 2.35·26-s − 0.384·27-s − 1.46·30-s − 1.43·31-s + 0.696·33-s − 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−19.4914718270, −18.9437167936, −18.7871956247, −18.0636542029, −17.4372274657, −17.1416936480, −16.7221615067, −16.1920174169, −15.6038578732, −14.8059546844, −14.2687617053, −13.0450433708, −12.9583864139, −11.7961223953, −11.7573247228, −10.7751381625, −9.93309835361, −9.81996681750, −9.03234585169, −8.01433080787, −7.59911177067, −6.87039121695, −5.44973416215, −5.00317001401, −3.50910294340, 0,
3.50910294340, 5.00317001401, 5.44973416215, 6.87039121695, 7.59911177067, 8.01433080787, 9.03234585169, 9.81996681750, 9.93309835361, 10.7751381625, 11.7573247228, 11.7961223953, 12.9583864139, 13.0450433708, 14.2687617053, 14.8059546844, 15.6038578732, 16.1920174169, 16.7221615067, 17.1416936480, 17.4372274657, 18.0636542029, 18.7871956247, 18.9437167936, 19.4914718270
