| L(s) = 1 | + 2·2-s − 3·3-s − 4·5-s − 6·6-s − 4·8-s + 3·9-s − 8·10-s + 4·11-s + 2·13-s + 12·15-s − 4·16-s − 3·17-s + 6·18-s − 3·19-s + 8·22-s + 12·24-s + 4·25-s + 4·26-s − 5·29-s + 24·30-s + 2·31-s − 12·33-s − 6·34-s − 37-s − 6·38-s − 6·39-s + 16·40-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s − 1.78·5-s − 2.44·6-s − 1.41·8-s + 9-s − 2.52·10-s + 1.20·11-s + 0.554·13-s + 3.09·15-s − 16-s − 0.727·17-s + 1.41·18-s − 0.688·19-s + 1.70·22-s + 2.44·24-s + 4/5·25-s + 0.784·26-s − 0.928·29-s + 4.38·30-s + 0.359·31-s − 2.08·33-s − 1.02·34-s − 0.164·37-s − 0.973·38-s − 0.960·39-s + 2.52·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820613 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820613 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 28297 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 125 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 59 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 53 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 180 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−12.4483399214, −11.9621172123, −11.7460511965, −11.4692793349, −11.2413835423, −10.8063389887, −10.3983755932, −9.69863458021, −9.18280693463, −8.89560732916, −8.33526466932, −8.01533827491, −7.33619358345, −6.96305660015, −6.30794621579, −6.16397393398, −5.69086654480, −5.16296186183, −4.71429253985, −4.29405275698, −3.96601315024, −3.66748025985, −3.07726580502, −1.90405430234, −0.695743644952, 0,
0.695743644952, 1.90405430234, 3.07726580502, 3.66748025985, 3.96601315024, 4.29405275698, 4.71429253985, 5.16296186183, 5.69086654480, 6.16397393398, 6.30794621579, 6.96305660015, 7.33619358345, 8.01533827491, 8.33526466932, 8.89560732916, 9.18280693463, 9.69863458021, 10.3983755932, 10.8063389887, 11.2413835423, 11.4692793349, 11.7460511965, 11.9621172123, 12.4483399214
