| L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 5·7-s − 3·8-s + 2·10-s − 3·11-s − 2·12-s + 8·13-s + 5·14-s + 4·15-s + 16-s + 2·17-s − 2·20-s + 10·21-s + 3·22-s + 6·24-s + 2·25-s − 8·26-s + 5·27-s − 5·28-s − 9·29-s − 4·30-s − 6·31-s + 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.88·7-s − 1.06·8-s + 0.632·10-s − 0.904·11-s − 0.577·12-s + 2.21·13-s + 1.33·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.447·20-s + 2.18·21-s + 0.639·22-s + 1.22·24-s + 2/5·25-s − 1.56·26-s + 0.962·27-s − 0.944·28-s − 1.67·29-s − 0.730·30-s − 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 94 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T - 44 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 58 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 146 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−18.4040714569, −18.0794695906, −17.3312437063, −16.6379842698, −16.2946067817, −16.0259289988, −15.5428728736, −15.1278323064, −14.1381260216, −13.1798821735, −12.9858409175, −12.3642423551, −11.5232576059, −11.4074963898, −10.6495638577, −10.2458738799, −9.14812865976, −9.00969015876, −8.06431244197, −7.28815946863, −6.36097611109, −6.11219655212, −5.39754514377, −3.72587974133, −3.15746762559, 0,
3.15746762559, 3.72587974133, 5.39754514377, 6.11219655212, 6.36097611109, 7.28815946863, 8.06431244197, 9.00969015876, 9.14812865976, 10.2458738799, 10.6495638577, 11.4074963898, 11.5232576059, 12.3642423551, 12.9858409175, 13.1798821735, 14.1381260216, 15.1278323064, 15.5428728736, 16.0259289988, 16.2946067817, 16.6379842698, 17.3312437063, 18.0794695906, 18.4040714569
