| L(s) = 1 | − 4·4-s + 36·13-s − 13·16-s − 48·19-s − 40·25-s + 40·31-s + 52·37-s + 140·43-s + 14·49-s − 144·52-s + 288·61-s + 104·64-s + 132·67-s − 188·73-s + 192·76-s − 76·79-s + 160·97-s + 160·100-s − 184·103-s + 328·109-s + 464·121-s − 160·124-s + 127-s + 131-s + 137-s + 139-s − 208·148-s + ⋯ |
| L(s) = 1 | − 4-s + 2.76·13-s − 0.812·16-s − 2.52·19-s − 8/5·25-s + 1.29·31-s + 1.40·37-s + 3.25·43-s + 2/7·49-s − 2.76·52-s + 4.72·61-s + 13/8·64-s + 1.97·67-s − 2.57·73-s + 2.52·76-s − 0.962·79-s + 1.64·97-s + 8/5·100-s − 1.78·103-s + 3.00·109-s + 3.83·121-s − 1.29·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.40·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.201219358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.201219358\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + p^{2} T^{2} + 29 T^{4} + p^{6} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 8 p T^{2} + 1307 T^{4} + 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 464 T^{2} + 83099 T^{4} - 464 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 18 T + 412 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 376 T^{2} + 80418 T^{4} - 376 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 24 T + 523 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1336 T^{2} + 854643 T^{4} - 1336 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1112 T^{2} + 1664450 T^{4} - 1112 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 20 T + 839 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 26 T - 13 p T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 5528 T^{2} + 13235771 T^{4} - 5528 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 70 T + 3348 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 4456 T^{2} + 10198434 T^{4} - 4456 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 9844 T^{2} + 39643158 T^{4} - 9844 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 3568 T^{2} + 635826 T^{4} - 3568 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 144 T + 11926 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 66 T + 6364 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 9616 T^{2} + 72347523 T^{4} - 9616 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 94 T + 12692 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 38 T + 11268 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 13984 T^{2} + 139005618 T^{4} - 13984 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 28304 T^{2} + 324704339 T^{4} - 28304 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 80 T + 17030 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.898147393268865166219460007823, −8.712922005625678938926573025888, −8.347952608025344022939181259992, −8.310782518013089284413698638405, −8.267307339652795593814267019001, −7.59914572549747203251965824758, −7.22430120390574426413990746844, −7.18851807216512888061728722536, −6.52410439288956774032694763610, −6.47630001991422402188450904474, −5.95181494010663835341818846769, −5.92803095851490694075688131557, −5.87100154760007751006762718250, −5.17795544687596938370658913589, −4.84653302238203726524711875246, −4.34240232919681319874182734348, −4.09686092634987802290807770820, −4.02936669833723122438795236836, −3.88309497078710048349157427112, −3.21017752843723406380439389958, −2.60293449140380309263588697661, −2.20247004354222222976786371787, −1.90906493121477313643922568113, −0.879714582107235026578769306463, −0.62882790953537701146558604373,
0.62882790953537701146558604373, 0.879714582107235026578769306463, 1.90906493121477313643922568113, 2.20247004354222222976786371787, 2.60293449140380309263588697661, 3.21017752843723406380439389958, 3.88309497078710048349157427112, 4.02936669833723122438795236836, 4.09686092634987802290807770820, 4.34240232919681319874182734348, 4.84653302238203726524711875246, 5.17795544687596938370658913589, 5.87100154760007751006762718250, 5.92803095851490694075688131557, 5.95181494010663835341818846769, 6.47630001991422402188450904474, 6.52410439288956774032694763610, 7.18851807216512888061728722536, 7.22430120390574426413990746844, 7.59914572549747203251965824758, 8.267307339652795593814267019001, 8.310782518013089284413698638405, 8.347952608025344022939181259992, 8.712922005625678938926573025888, 8.898147393268865166219460007823
