| L(s) = 1 | − 4·2-s + 2·3-s + 8·4-s + 4·5-s − 8·6-s − 8·8-s + 3·9-s − 16·10-s + 16·12-s + 8·15-s − 4·16-s − 12·17-s − 12·18-s + 32·20-s − 16·24-s + 11·25-s + 4·27-s − 4·29-s − 32·30-s + 32·32-s + 48·34-s + 24·36-s − 2·37-s − 32·40-s − 2·43-s + 12·45-s + 16·47-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.15·3-s + 4·4-s + 1.78·5-s − 3.26·6-s − 2.82·8-s + 9-s − 5.05·10-s + 4.61·12-s + 2.06·15-s − 16-s − 2.91·17-s − 2.82·18-s + 7.15·20-s − 3.26·24-s + 11/5·25-s + 0.769·27-s − 0.742·29-s − 5.84·30-s + 5.65·32-s + 8.23·34-s + 4·36-s − 0.328·37-s − 5.05·40-s − 0.304·43-s + 1.78·45-s + 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7695478471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7695478471\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−10.97354914286742907840547048921, −10.64376853280538732873835196580, −10.16611440680651221428335497561, −9.825559453686683173643219770104, −9.307911693400769796865292465243, −9.158389456161346471318345142439, −8.875556592568695344666217360988, −8.435083951764939988995447218328, −8.095690002938542576854341536020, −7.38245951446731992835070226902, −6.82866517287244109916693696737, −6.79568999643739799447282039592, −6.09095042061882719816334988417, −5.15598333046146503472668996480, −4.57114576488874229321384665906, −3.85144277930253994974000303287, −2.52509541441722889404633038325, −2.11783780223542915033328418052, −1.96327910702602339931425882472, −0.834978617783562150213868522314,
0.834978617783562150213868522314, 1.96327910702602339931425882472, 2.11783780223542915033328418052, 2.52509541441722889404633038325, 3.85144277930253994974000303287, 4.57114576488874229321384665906, 5.15598333046146503472668996480, 6.09095042061882719816334988417, 6.79568999643739799447282039592, 6.82866517287244109916693696737, 7.38245951446731992835070226902, 8.095690002938542576854341536020, 8.435083951764939988995447218328, 8.875556592568695344666217360988, 9.158389456161346471318345142439, 9.307911693400769796865292465243, 9.825559453686683173643219770104, 10.16611440680651221428335497561, 10.64376853280538732873835196580, 10.97354914286742907840547048921
