L(s) = 1 | − 2·5-s + 9-s + 25-s − 2·29-s − 2·41-s − 2·45-s − 49-s + 2·61-s − 4·89-s + 4·101-s − 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 2·5-s + 9-s + 25-s − 2·29-s − 2·41-s − 2·45-s − 49-s + 2·61-s − 4·89-s + 4·101-s − 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2717722466\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2717722466\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.33438963666772706290111264723, −6.27479980721362033906995617869, −6.13955074586187396378085917121, −5.58718804853802737488813131175, −5.53277296425893229032705217626, −5.51988635997385261160774587765, −5.12352981424206365054936397851, −5.01900748094432207229325186479, −4.67820127715461474887788639533, −4.47822221018584133888104016738, −4.23659397101326177822823679224, −4.09940668331711756428119696595, −3.92548777695194507721642050241, −3.79362776807097909848194366055, −3.42716770290077940993653432234, −3.27236108783644115050752032575, −3.24754689870159944293772371394, −2.66862849851516393678702060796, −2.62351533429455454109864376841, −1.99795644573682382093650688303, −1.97488064813586656689865332117, −1.70702938926254345453074260019, −1.22300475139456811455549182232, −1.05167309494442332897635701643, −0.23598449708717053306657858830,
0.23598449708717053306657858830, 1.05167309494442332897635701643, 1.22300475139456811455549182232, 1.70702938926254345453074260019, 1.97488064813586656689865332117, 1.99795644573682382093650688303, 2.62351533429455454109864376841, 2.66862849851516393678702060796, 3.24754689870159944293772371394, 3.27236108783644115050752032575, 3.42716770290077940993653432234, 3.79362776807097909848194366055, 3.92548777695194507721642050241, 4.09940668331711756428119696595, 4.23659397101326177822823679224, 4.47822221018584133888104016738, 4.67820127715461474887788639533, 5.01900748094432207229325186479, 5.12352981424206365054936397851, 5.51988635997385261160774587765, 5.53277296425893229032705217626, 5.58718804853802737488813131175, 6.13955074586187396378085917121, 6.27479980721362033906995617869, 6.33438963666772706290111264723