| L(s) = 1 | − 3-s + 4-s + 7-s + 9-s − 12-s − 4·13-s − 3·16-s + 3·19-s − 21-s + 2·25-s − 27-s + 28-s + 36-s − 12·37-s + 4·39-s − 8·43-s + 3·48-s − 6·49-s − 4·52-s − 3·57-s + 20·61-s + 63-s − 7·64-s − 8·67-s + 4·73-s − 2·75-s + 3·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 3/4·16-s + 0.688·19-s − 0.218·21-s + 2/5·25-s − 0.192·27-s + 0.188·28-s + 1/6·36-s − 1.97·37-s + 0.640·39-s − 1.21·43-s + 0.433·48-s − 6/7·49-s − 0.554·52-s − 0.397·57-s + 2.56·61-s + 0.125·63-s − 7/8·64-s − 0.977·67-s + 0.468·73-s − 0.230·75-s + 0.344·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3591 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3591 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7175329072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7175329072\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−12.40455669710029462914444723137, −11.95540212762440671293789634833, −11.50241564803008008621431292804, −10.96209673656104862541748065987, −10.23323249431693370205575833354, −9.799114190252748412073333481185, −8.962359067748700101158432842733, −8.271522425507048577984329398951, −7.32472320975423953442563815704, −7.00282080312508626852911634054, −6.20992784451237885601763112834, −5.15770247927932104238955318180, −4.79548037719004453433730782907, −3.43778539088294327646502733438, −2.08672222628823078726316203122,
2.08672222628823078726316203122, 3.43778539088294327646502733438, 4.79548037719004453433730782907, 5.15770247927932104238955318180, 6.20992784451237885601763112834, 7.00282080312508626852911634054, 7.32472320975423953442563815704, 8.271522425507048577984329398951, 8.962359067748700101158432842733, 9.799114190252748412073333481185, 10.23323249431693370205575833354, 10.96209673656104862541748065987, 11.50241564803008008621431292804, 11.95540212762440671293789634833, 12.40455669710029462914444723137
