| L(s) = 1 | − 4-s − 2·7-s − 2·9-s + 11-s + 13-s − 3·16-s + 3·17-s − 6·19-s + 2·25-s − 3·27-s + 2·28-s + 8·29-s + 5·31-s + 2·36-s + 7·37-s + 5·41-s − 13·43-s − 44-s + 12·47-s − 2·49-s − 52-s − 10·53-s − 16·59-s − 4·61-s + 4·63-s + 7·64-s + 2·67-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s − 3/4·16-s + 0.727·17-s − 1.37·19-s + 2/5·25-s − 0.577·27-s + 0.377·28-s + 1.48·29-s + 0.898·31-s + 1/3·36-s + 1.15·37-s + 0.780·41-s − 1.98·43-s − 0.150·44-s + 1.75·47-s − 2/7·49-s − 0.138·52-s − 1.37·53-s − 2.08·59-s − 0.512·61-s + 0.503·63-s + 7/8·64-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1473 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1473 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5120766201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5120766201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 491 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 36 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 190 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 130 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 152 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−19.2353345938, −18.8305742964, −18.4149405839, −17.5516695065, −17.2586259214, −16.6445902685, −16.1157230183, −15.4579459120, −14.8701154857, −14.2426067664, −13.6270709711, −13.2668853218, −12.3534843155, −12.0784139881, −11.0654580980, −10.6230904582, −9.66134133115, −9.25256332363, −8.44025798781, −7.88132777215, −6.50293306524, −6.30963389390, −5.02040793865, −4.08732928883, −2.84377911189,
2.84377911189, 4.08732928883, 5.02040793865, 6.30963389390, 6.50293306524, 7.88132777215, 8.44025798781, 9.25256332363, 9.66134133115, 10.6230904582, 11.0654580980, 12.0784139881, 12.3534843155, 13.2668853218, 13.6270709711, 14.2426067664, 14.8701154857, 15.4579459120, 16.1157230183, 16.6445902685, 17.2586259214, 17.5516695065, 18.4149405839, 18.8305742964, 19.2353345938
