| L(s) = 1 | + 2-s − 2·3-s − 4-s − 3·5-s − 2·6-s − 3·7-s − 3·8-s − 9-s − 3·10-s − 5·11-s + 2·12-s − 6·13-s − 3·14-s + 6·15-s − 16-s + 4·17-s − 18-s + 2·19-s + 3·20-s + 6·21-s − 5·22-s + 4·23-s + 6·24-s + 4·25-s − 6·26-s + 6·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.34·5-s − 0.816·6-s − 1.13·7-s − 1.06·8-s − 1/3·9-s − 0.948·10-s − 1.50·11-s + 0.577·12-s − 1.66·13-s − 0.801·14-s + 1.54·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.670·20-s + 1.30·21-s − 1.06·22-s + 0.834·23-s + 1.22·24-s + 4/5·25-s − 1.17·26-s + 1.15·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 15731 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 20 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 94 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T - 110 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 149 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 141 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
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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
−15.1026005818, −14.5484128513, −14.1506352966, −13.3535311808, −13.0250896047, −12.7104419241, −12.2933292133, −11.8539229833, −11.5766043259, −10.9785423621, −10.5830310388, −9.91857985876, −9.43868224274, −9.11518248559, −8.20903902363, −7.74452962880, −7.34120050864, −6.81318256726, −5.83734188488, −5.60904842215, −5.11372320246, −4.69536883732, −3.70208618124, −3.27931863224, −2.65740420242, 0, 0,
2.65740420242, 3.27931863224, 3.70208618124, 4.69536883732, 5.11372320246, 5.60904842215, 5.83734188488, 6.81318256726, 7.34120050864, 7.74452962880, 8.20903902363, 9.11518248559, 9.43868224274, 9.91857985876, 10.5830310388, 10.9785423621, 11.5766043259, 11.8539229833, 12.2933292133, 12.7104419241, 13.0250896047, 13.3535311808, 14.1506352966, 14.5484128513, 15.1026005818
