| L(s) = 1 | − 2·2-s − 3·3-s − 5·5-s + 6·6-s − 4·7-s + 4·8-s + 4·9-s + 10·10-s + 11-s − 6·13-s + 8·14-s + 15·15-s − 4·16-s − 2·17-s − 8·18-s + 19-s + 12·21-s − 2·22-s − 4·23-s − 12·24-s + 12·25-s + 12·26-s − 6·27-s − 30·30-s − 3·31-s − 3·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 2.23·5-s + 2.44·6-s − 1.51·7-s + 1.41·8-s + 4/3·9-s + 3.16·10-s + 0.301·11-s − 1.66·13-s + 2.13·14-s + 3.87·15-s − 16-s − 0.485·17-s − 1.88·18-s + 0.229·19-s + 2.61·21-s − 0.426·22-s − 0.834·23-s − 2.44·24-s + 12/5·25-s + 2.35·26-s − 1.15·27-s − 5.47·30-s − 0.538·31-s − 0.522·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 26 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 88 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 67 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T - 41 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 56 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−17.8622113402, −17.2224859645, −16.8034686469, −16.5471475014, −16.0652302914, −15.6296716220, −15.1030466095, −14.4193709862, −13.4814110772, −12.8017874579, −12.3790303901, −11.8286472719, −11.5848168181, −10.9363249564, −10.2395446949, −9.77782737143, −9.31337709479, −8.52730813457, −7.82884772682, −7.35343795733, −6.78745693078, −5.94162651486, −4.89248014557, −4.31867894568, −3.42810206293, 0, 0,
3.42810206293, 4.31867894568, 4.89248014557, 5.94162651486, 6.78745693078, 7.35343795733, 7.82884772682, 8.52730813457, 9.31337709479, 9.77782737143, 10.2395446949, 10.9363249564, 11.5848168181, 11.8286472719, 12.3790303901, 12.8017874579, 13.4814110772, 14.4193709862, 15.1030466095, 15.6296716220, 16.0652302914, 16.5471475014, 16.8034686469, 17.2224859645, 17.8622113402
