| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 7-s + 2·8-s + 4·9-s + 6·10-s − 6·11-s − 3·12-s + 3·13-s + 2·14-s + 9·15-s − 3·16-s − 8·18-s − 19-s − 3·20-s + 3·21-s + 12·22-s + 6·23-s − 6·24-s + 3·25-s − 6·26-s − 6·27-s − 28-s − 7·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 0.377·7-s + 0.707·8-s + 4/3·9-s + 1.89·10-s − 1.80·11-s − 0.866·12-s + 0.832·13-s + 0.534·14-s + 2.32·15-s − 3/4·16-s − 1.88·18-s − 0.229·19-s − 0.670·20-s + 0.654·21-s + 2.55·22-s + 1.25·23-s − 1.22·24-s + 3/5·25-s − 1.17·26-s − 1.15·27-s − 0.188·28-s − 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1094 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1094 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 547 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 19 T + p T^{2} ) \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 52 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 49 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 147 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 122 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−19.9940553546, −19.1427323879, −18.7401343512, −18.5210511443, −17.9938709566, −17.3088737039, −16.9230026700, −16.2241703191, −16.1342690933, −15.3477434207, −14.8444330983, −13.2898718967, −13.1232785391, −12.3314888688, −11.4802384117, −11.0581471797, −10.6881184346, −10.0274648633, −9.04391609052, −8.36870753049, −7.53056213590, −7.12598095792, −5.73754904886, −5.16164216987, −3.80451791311, 0,
3.80451791311, 5.16164216987, 5.73754904886, 7.12598095792, 7.53056213590, 8.36870753049, 9.04391609052, 10.0274648633, 10.6881184346, 11.0581471797, 11.4802384117, 12.3314888688, 13.1232785391, 13.2898718967, 14.8444330983, 15.3477434207, 16.1342690933, 16.2241703191, 16.9230026700, 17.3088737039, 17.9938709566, 18.5210511443, 18.7401343512, 19.1427323879, 19.9940553546
