| L(s) = 1 | + 2·4-s − 2·7-s − 5·13-s + 3·19-s + 2·25-s − 4·28-s − 17·31-s + 13·37-s + 10·43-s − 2·49-s − 10·52-s + 4·61-s − 8·64-s + 13·67-s − 5·73-s + 6·76-s − 17·79-s + 10·91-s + 97-s + 4·100-s + 103-s + 13·109-s − 4·121-s − 34·124-s + 127-s + 131-s − 6·133-s + ⋯ |
| L(s) = 1 | + 4-s − 0.755·7-s − 1.38·13-s + 0.688·19-s + 2/5·25-s − 0.755·28-s − 3.05·31-s + 2.13·37-s + 1.52·43-s − 2/7·49-s − 1.38·52-s + 0.512·61-s − 64-s + 1.58·67-s − 0.585·73-s + 0.688·76-s − 1.91·79-s + 1.04·91-s + 0.101·97-s + 2/5·100-s + 0.0985·103-s + 1.24·109-s − 0.363·121-s − 3.05·124-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4617 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4617 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8952962965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8952962965\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 173 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−12.43427899284152918546021258704, −11.63976431820768373888186441453, −11.22902937833253117763654423309, −10.71587933259997510037831955292, −9.878308348368498170316967181471, −9.477534546797145634195949660686, −8.912565401482021351704846718260, −7.74868034272598788486147843554, −7.32865285631809519045402399945, −6.85236248478523798277613960203, −5.98210814456275682109313669149, −5.33838939032273262041580322110, −4.24450362066822217357110333143, −3.10903483945463787321826338399, −2.24426118984549632904266470806,
2.24426118984549632904266470806, 3.10903483945463787321826338399, 4.24450362066822217357110333143, 5.33838939032273262041580322110, 5.98210814456275682109313669149, 6.85236248478523798277613960203, 7.32865285631809519045402399945, 7.74868034272598788486147843554, 8.912565401482021351704846718260, 9.477534546797145634195949660686, 9.878308348368498170316967181471, 10.71587933259997510037831955292, 11.22902937833253117763654423309, 11.63976431820768373888186441453, 12.43427899284152918546021258704
