L(s) = 1 | − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s − 8·13-s + 16-s + 4·19-s − 4·21-s − 10·25-s + 4·27-s + 2·28-s − 8·31-s + 36-s + 4·37-s + 16·39-s + 16·43-s − 2·48-s + 3·49-s − 8·52-s − 8·57-s + 16·61-s + 2·63-s + 64-s − 8·67-s + 4·73-s + 20·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 0.872·21-s − 2·25-s + 0.769·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s + 0.657·37-s + 2.56·39-s + 2.43·43-s − 0.288·48-s + 3/7·49-s − 1.10·52-s − 1.05·57-s + 2.04·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 2.30·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5054223186\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5054223186\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−13.45852272915611541668421078683, −12.37171571172372853840948077604, −12.30526099005271094386573698884, −11.52629384186915375545261187858, −11.23136141438460806904855801814, −10.47027853370113893616936981615, −9.765547119459919407856461234632, −9.207333736191082013174831123730, −7.80365099617056766338064266882, −7.57571100088867902110310233811, −6.71310127772875365703449938682, −5.57928681742950427486583645839, −5.31921059379060528260176389250, −4.19102102405251164216473944262, −2.39101862347891857041372548490,
2.39101862347891857041372548490, 4.19102102405251164216473944262, 5.31921059379060528260176389250, 5.57928681742950427486583645839, 6.71310127772875365703449938682, 7.57571100088867902110310233811, 7.80365099617056766338064266882, 9.207333736191082013174831123730, 9.765547119459919407856461234632, 10.47027853370113893616936981615, 11.23136141438460806904855801814, 11.52629384186915375545261187858, 12.30526099005271094386573698884, 12.37171571172372853840948077604, 13.45852272915611541668421078683