L(s) = 1 | + 2·4-s + 4·11-s + 3·16-s − 20·19-s + 16·29-s + 4·41-s + 8·44-s + 22·49-s + 44·59-s + 24·61-s + 12·64-s − 20·71-s − 40·76-s − 36·79-s − 8·89-s − 20·101-s + 24·109-s + 32·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 4-s + 1.20·11-s + 3/4·16-s − 4.58·19-s + 2.97·29-s + 0.624·41-s + 1.20·44-s + 22/7·49-s + 5.72·59-s + 3.07·61-s + 3/2·64-s − 2.37·71-s − 4.58·76-s − 4.05·79-s − 0.847·89-s − 1.99·101-s + 2.29·109-s + 2.97·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.169405614\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.169405614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1531 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 114 T^{2} + 5699 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 36 T^{2} - 586 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 42 T^{2} + 4571 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8726 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 10 T + 159 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 19910 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 18 T + 221 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 15254 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 354 T^{2} + 49859 T^{4} - 354 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.44295544707527440008838545362, −6.23191696830421030612315867783, −5.91941450071104346119706323586, −5.83864675941068292794131522268, −5.52153634804185277992862589116, −5.47793647540764574041079792548, −5.01424335622174041718690737792, −5.01311473730984656468624379926, −4.36931673810985862742300796462, −4.34850907095944858481120054710, −4.13315223288798006548651766528, −4.05294231511408836867187843073, −4.00282648596314592558570647686, −3.74418375022745920690178854001, −3.09824448593868953450048168934, −2.95275360860394555886894667332, −2.85454320349412162939625092790, −2.32890239594830230500865583367, −2.17016719000257800325306429748, −2.15509539755876009654417942676, −2.07220445527602218971514067218, −1.26639940084778147362999521422, −1.12078926378051411794122827619, −0.825837566133919069551858940146, −0.36380217726064410013608527407,
0.36380217726064410013608527407, 0.825837566133919069551858940146, 1.12078926378051411794122827619, 1.26639940084778147362999521422, 2.07220445527602218971514067218, 2.15509539755876009654417942676, 2.17016719000257800325306429748, 2.32890239594830230500865583367, 2.85454320349412162939625092790, 2.95275360860394555886894667332, 3.09824448593868953450048168934, 3.74418375022745920690178854001, 4.00282648596314592558570647686, 4.05294231511408836867187843073, 4.13315223288798006548651766528, 4.34850907095944858481120054710, 4.36931673810985862742300796462, 5.01311473730984656468624379926, 5.01424335622174041718690737792, 5.47793647540764574041079792548, 5.52153634804185277992862589116, 5.83864675941068292794131522268, 5.91941450071104346119706323586, 6.23191696830421030612315867783, 6.44295544707527440008838545362