L(s) = 1 | + 2·3-s + 4-s + 3·5-s + 7-s + 2·12-s + 6·15-s + 16-s + 3·20-s + 2·21-s − 25-s − 5·27-s + 28-s + 9·29-s + 31-s + 3·35-s + 4·43-s + 2·48-s − 11·49-s + 6·60-s + 13·61-s + 64-s − 9·71-s − 2·73-s − 2·75-s + 3·80-s − 4·81-s − 18·83-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.577·12-s + 1.54·15-s + 1/4·16-s + 0.670·20-s + 0.436·21-s − 1/5·25-s − 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.179·31-s + 0.507·35-s + 0.609·43-s + 0.288·48-s − 1.57·49-s + 0.774·60-s + 1.66·61-s + 1/8·64-s − 1.06·71-s − 0.234·73-s − 0.230·75-s + 0.335·80-s − 4/9·81-s − 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.463191633\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.463191633\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 109 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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
−9.336473098750040850489155655124, −8.806840308673572560730737074186, −8.391899483979675592482755086185, −7.966666320516702555547746942422, −7.54411649259926653044801358487, −6.72286979169083995428434301157, −6.46644206722297564797061396645, −5.75530435121550658840676090845, −5.44730954160383960785821623639, −4.68128576699473337277481189035, −3.99512304643175620632381373108, −3.16487986384095514047126023359, −2.65616251387774940163246897532, −2.13446785778834027600131184348, −1.40436374869107075146223341051,
1.40436374869107075146223341051, 2.13446785778834027600131184348, 2.65616251387774940163246897532, 3.16487986384095514047126023359, 3.99512304643175620632381373108, 4.68128576699473337277481189035, 5.44730954160383960785821623639, 5.75530435121550658840676090845, 6.46644206722297564797061396645, 6.72286979169083995428434301157, 7.54411649259926653044801358487, 7.966666320516702555547746942422, 8.391899483979675592482755086185, 8.806840308673572560730737074186, 9.336473098750040850489155655124