| L(s) = 1 | − 4·5-s + 9-s − 4·13-s − 16·23-s + 2·25-s + 6·29-s − 4·45-s − 14·49-s − 4·53-s + 8·59-s + 16·65-s − 8·67-s + 16·71-s + 81-s − 8·83-s + 32·103-s − 24·107-s − 4·109-s + 64·115-s − 4·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s − 1.10·13-s − 3.33·23-s + 2/5·25-s + 1.11·29-s − 0.596·45-s − 2·49-s − 0.549·53-s + 1.04·59-s + 1.98·65-s − 0.977·67-s + 1.89·71-s + 1/9·81-s − 0.878·83-s + 3.15·103-s − 2.32·107-s − 0.383·109-s + 5.96·115-s − 0.369·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4317936316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4317936316\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−8.169834685173383924757119979209, −8.098990694093691505068092710868, −7.71721200903054213816393191968, −7.42501431214799645944696043073, −6.64608831177428132224128205181, −6.42897107072744719896577758454, −5.74261611440312941575828325994, −5.15764774786080872178121149571, −4.46164747667887486388909533877, −4.25303028692796488061253338187, −3.74922469853547417385089284767, −3.23320358630252677719161660912, −2.37294675688448229559246749927, −1.75758602678204178340122440060, −0.33671766043668980558621258503,
0.33671766043668980558621258503, 1.75758602678204178340122440060, 2.37294675688448229559246749927, 3.23320358630252677719161660912, 3.74922469853547417385089284767, 4.25303028692796488061253338187, 4.46164747667887486388909533877, 5.15764774786080872178121149571, 5.74261611440312941575828325994, 6.42897107072744719896577758454, 6.64608831177428132224128205181, 7.42501431214799645944696043073, 7.71721200903054213816393191968, 8.098990694093691505068092710868, 8.169834685173383924757119979209
