| L(s) = 1 | + 2·3-s + 2·7-s − 6·11-s − 4·13-s − 2·17-s + 4·19-s + 4·21-s + 6·23-s + 2·25-s − 2·27-s + 2·31-s − 12·33-s − 16·37-s − 8·39-s − 12·41-s + 4·43-s − 8·49-s − 4·51-s − 12·53-s + 8·57-s + 12·59-s + 8·61-s + 16·67-s + 12·69-s + 6·71-s + 4·73-s + 4·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s − 1.80·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 2/5·25-s − 0.384·27-s + 0.359·31-s − 2.08·33-s − 2.63·37-s − 1.28·39-s − 1.87·41-s + 0.609·43-s − 8/7·49-s − 0.560·51-s − 1.64·53-s + 1.05·57-s + 1.56·59-s + 1.02·61-s + 1.95·67-s + 1.44·69-s + 0.712·71-s + 0.468·73-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.319698624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319698624\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 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 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.06568055112947454578800609571, −9.711697319825631725390397653390, −9.093719552804831514940715237480, −8.770908234888405638304989285304, −8.321619323640742835079424426267, −8.220642794620788386503302674655, −7.56366648465131126711685281820, −7.44599551055561857195532579541, −6.71303151046301222463820761184, −6.62249290870410303000612414614, −5.40275666838476630385164389814, −5.36199608195030022707353505965, −4.80362808321825323764039736383, −4.76722512150068504889724608070, −3.49952373717517545902107802904, −3.34189242308796861147881147307, −2.82916408279423896461630018632, −2.17437638900435866349219462895, −1.90853919242954552946609169359, −0.60223030651800219820213822827,
0.60223030651800219820213822827, 1.90853919242954552946609169359, 2.17437638900435866349219462895, 2.82916408279423896461630018632, 3.34189242308796861147881147307, 3.49952373717517545902107802904, 4.76722512150068504889724608070, 4.80362808321825323764039736383, 5.36199608195030022707353505965, 5.40275666838476630385164389814, 6.62249290870410303000612414614, 6.71303151046301222463820761184, 7.44599551055561857195532579541, 7.56366648465131126711685281820, 8.220642794620788386503302674655, 8.321619323640742835079424426267, 8.770908234888405638304989285304, 9.093719552804831514940715237480, 9.711697319825631725390397653390, 10.06568055112947454578800609571
