| L(s) = 1 | − 3-s − 4-s + 2·5-s + 3·9-s − 11-s + 12-s − 13-s − 2·15-s − 3·16-s + 2·17-s + 4·19-s − 2·20-s − 3·23-s − 8·27-s − 6·29-s + 33-s − 3·36-s − 2·37-s + 39-s + 4·41-s − 7·43-s + 44-s + 6·45-s + 3·47-s + 3·48-s − 6·49-s − 2·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.894·5-s + 9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.516·15-s − 3/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.625·23-s − 1.53·27-s − 1.11·29-s + 0.174·33-s − 1/2·36-s − 0.328·37-s + 0.160·39-s + 0.624·41-s − 1.06·43-s + 0.150·44-s + 0.894·45-s + 0.437·47-s + 0.433·48-s − 6/7·49-s − 0.280·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6746680557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6746680557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 503 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 42 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 74 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 128 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 88 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−17.9541397667, −17.4922818404, −17.1941750656, −16.3433830243, −16.1279754200, −15.4382468507, −14.8766658342, −14.1159738519, −13.7316071760, −13.1678464585, −12.7968779149, −12.0582462866, −11.4585510551, −10.8748825910, −10.0287722428, −9.67768406969, −9.32197959014, −8.28573067136, −7.51879883325, −6.90545247992, −5.96802899229, −5.40027617984, −4.65934329096, −3.63215235605, −1.97580302488,
1.97580302488, 3.63215235605, 4.65934329096, 5.40027617984, 5.96802899229, 6.90545247992, 7.51879883325, 8.28573067136, 9.32197959014, 9.67768406969, 10.0287722428, 10.8748825910, 11.4585510551, 12.0582462866, 12.7968779149, 13.1678464585, 13.7316071760, 14.1159738519, 14.8766658342, 15.4382468507, 16.1279754200, 16.3433830243, 17.1941750656, 17.4922818404, 17.9541397667
