| L(s) = 1 | + 2-s + 2·3-s + 2·6-s + 8-s − 8·11-s − 16-s + 4·17-s − 8·22-s + 2·23-s + 2·24-s + 2·25-s − 2·27-s + 29-s − 4·31-s − 6·32-s − 16·33-s + 4·34-s + 37-s + 2·46-s + 6·47-s − 2·48-s − 4·49-s + 2·50-s + 8·51-s + 8·53-s − 2·54-s + 58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.816·6-s + 0.353·8-s − 2.41·11-s − 1/4·16-s + 0.970·17-s − 1.70·22-s + 0.417·23-s + 0.408·24-s + 2/5·25-s − 0.384·27-s + 0.185·29-s − 0.718·31-s − 1.06·32-s − 2.78·33-s + 0.685·34-s + 0.164·37-s + 0.294·46-s + 0.875·47-s − 0.288·48-s − 4/7·49-s + 0.282·50-s + 1.12·51-s + 1.09·53-s − 0.272·54-s + 0.131·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643918135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643918135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 9161 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 114 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 47 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−16.3883307480, −16.2022896707, −15.5557231334, −15.1128890456, −14.4839093106, −14.3550848304, −13.5235227529, −13.3890650535, −12.9517713795, −12.3961039287, −11.7154309001, −10.9001034244, −10.5154033694, −10.0589270481, −9.15652684872, −8.75284410941, −8.01738800790, −7.66533259030, −7.11158139402, −5.95615165267, −5.22878981030, −4.86190729176, −3.75270502591, −2.95823434535, −2.33951702631,
2.33951702631, 2.95823434535, 3.75270502591, 4.86190729176, 5.22878981030, 5.95615165267, 7.11158139402, 7.66533259030, 8.01738800790, 8.75284410941, 9.15652684872, 10.0589270481, 10.5154033694, 10.9001034244, 11.7154309001, 12.3961039287, 12.9517713795, 13.3890650535, 13.5235227529, 14.3550848304, 14.4839093106, 15.1128890456, 15.5557231334, 16.2022896707, 16.3883307480
