| L(s) = 1 | + 3·5-s + 6·7-s − 6·11-s − 6·17-s + 5·25-s + 18·35-s − 24·43-s + 11·49-s + 12·53-s − 18·55-s − 12·59-s + 22·61-s − 36·67-s − 36·77-s − 18·85-s + 22·109-s + 12·113-s − 36·119-s − 5·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 2.26·7-s − 1.80·11-s − 1.45·17-s + 25-s + 3.04·35-s − 3.65·43-s + 11/7·49-s + 1.64·53-s − 2.42·55-s − 1.56·59-s + 2.81·61-s − 4.39·67-s − 4.10·77-s − 1.95·85-s + 2.10·109-s + 1.12·113-s − 3.30·119-s − 0.454·121-s + 1.60·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1666121289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1666121289\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 804 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 1404 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 894 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 4110 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 155 T^{2} + 15996 T^{4} + 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 253 T^{2} + 27816 T^{4} - 253 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 12051 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 217 T^{2} + 24576 T^{4} - 217 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.55650806495351313734330942442, −6.29791518554838700581142263768, −5.92765666180913783255857966723, −5.68763712172091630465819991104, −5.55911503076568408229148657888, −5.27791652111663599452260034922, −5.20545765431737643776970111010, −5.16168136458176994172214236431, −4.63711602582515557834550545157, −4.57403377242500290524634475246, −4.54574875497110430236640190290, −4.34007414288102542843593549490, −3.80453996497509375913125544275, −3.63961415238653438065193388584, −3.30237613774999641583700878538, −3.03868922683481681583660990828, −2.70608988771403391409580771571, −2.57346707823982290295568572036, −2.27393495772494767115612513823, −1.95189971263426707987919967300, −1.85892783930051258060930749163, −1.51585885603914855031704530891, −1.36489634310748130748632957737, −0.856973222372948793086707018145, −0.05933950576915367796153963930,
0.05933950576915367796153963930, 0.856973222372948793086707018145, 1.36489634310748130748632957737, 1.51585885603914855031704530891, 1.85892783930051258060930749163, 1.95189971263426707987919967300, 2.27393495772494767115612513823, 2.57346707823982290295568572036, 2.70608988771403391409580771571, 3.03868922683481681583660990828, 3.30237613774999641583700878538, 3.63961415238653438065193388584, 3.80453996497509375913125544275, 4.34007414288102542843593549490, 4.54574875497110430236640190290, 4.57403377242500290524634475246, 4.63711602582515557834550545157, 5.16168136458176994172214236431, 5.20545765431737643776970111010, 5.27791652111663599452260034922, 5.55911503076568408229148657888, 5.68763712172091630465819991104, 5.92765666180913783255857966723, 6.29791518554838700581142263768, 6.55650806495351313734330942442
