| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 7-s + 8-s + 9-s − 2·12-s − 8·13-s + 14-s + 16-s − 12·17-s + 18-s + 4·19-s − 2·21-s − 2·24-s − 10·25-s − 8·26-s + 4·27-s + 28-s + 12·29-s + 32-s − 12·34-s + 36-s + 4·38-s + 16·39-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 0.267·14-s + 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s − 0.408·24-s − 2·25-s − 1.56·26-s + 0.769·27-s + 0.188·28-s + 2.22·29-s + 0.176·32-s − 2.05·34-s + 1/6·36-s + 0.648·38-s + 2.56·39-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084398903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084398903\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.678731346329368257431074943309, −7.895164678888831912291936607752, −7.80365099617056766338064266882, −6.90531576344093792893651903833, −6.71310127772875365703449938682, −6.50364338668347614528670710226, −5.58920639964454334454555282661, −5.31921059379060528260176389250, −4.91171613213171916096600238802, −4.29390759056738033972615396659, −4.19102102405251164216473944262, −2.99977174751051762376143721143, −2.39101862347891857041372548490, −1.99280794987984862469967531446, −0.51891811197874657341665639572,
0.51891811197874657341665639572, 1.99280794987984862469967531446, 2.39101862347891857041372548490, 2.99977174751051762376143721143, 4.19102102405251164216473944262, 4.29390759056738033972615396659, 4.91171613213171916096600238802, 5.31921059379060528260176389250, 5.58920639964454334454555282661, 6.50364338668347614528670710226, 6.71310127772875365703449938682, 6.90531576344093792893651903833, 7.80365099617056766338064266882, 7.895164678888831912291936607752, 8.678731346329368257431074943309
