| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 3·7-s + 2·8-s + 6·9-s + 6·10-s − 3·12-s − 2·13-s + 6·14-s + 9·15-s − 3·16-s − 2·17-s − 12·18-s − 3·19-s − 3·20-s + 9·21-s − 2·23-s − 6·24-s + 3·25-s + 4·26-s − 9·27-s − 3·28-s + 3·29-s − 18·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 1.13·7-s + 0.707·8-s + 2·9-s + 1.89·10-s − 0.866·12-s − 0.554·13-s + 1.60·14-s + 2.32·15-s − 3/4·16-s − 0.485·17-s − 2.82·18-s − 0.688·19-s − 0.670·20-s + 1.96·21-s − 0.417·23-s − 1.22·24-s + 3/5·25-s + 0.784·26-s − 1.73·27-s − 0.566·28-s + 0.557·29-s − 3.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1062 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1062 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 102 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−19.8772963860, −19.4043153922, −18.9526513704, −18.3802949176, −18.0738157481, −17.2970676600, −16.9062141799, −16.4405772503, −16.0364442060, −15.4755751303, −14.8240037819, −13.6175897810, −12.7762377015, −12.4225804689, −11.7565585421, −11.0462718272, −10.6211580847, −9.86441395466, −9.35258677494, −8.35151542537, −7.59724861641, −6.89663701974, −6.14411969783, −4.89186133195, −3.93517256570, 0,
3.93517256570, 4.89186133195, 6.14411969783, 6.89663701974, 7.59724861641, 8.35151542537, 9.35258677494, 9.86441395466, 10.6211580847, 11.0462718272, 11.7565585421, 12.4225804689, 12.7762377015, 13.6175897810, 14.8240037819, 15.4755751303, 16.0364442060, 16.4405772503, 16.9062141799, 17.2970676600, 18.0738157481, 18.3802949176, 18.9526513704, 19.4043153922, 19.8772963860
