| L(s) = 1 | + 4·3-s − 8·7-s + 4·9-s + 12·19-s − 32·21-s + 12·25-s − 4·27-s + 16·37-s + 34·49-s + 48·57-s + 36·59-s − 32·63-s + 48·75-s − 10·81-s + 36·83-s − 48·103-s + 64·111-s − 24·113-s + 12·121-s + 127-s + 131-s − 96·133-s + 137-s + 139-s + 136·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.02·7-s + 4/3·9-s + 2.75·19-s − 6.98·21-s + 12/5·25-s − 0.769·27-s + 2.63·37-s + 34/7·49-s + 6.35·57-s + 4.68·59-s − 4.03·63-s + 5.54·75-s − 1.11·81-s + 3.95·83-s − 4.72·103-s + 6.07·111-s − 2.25·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s − 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 11.2·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.949001578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.949001578\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 426 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 6186 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 18 T + 244 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.60710582639251470740998581017, −6.56508512973980881960217348007, −6.29759640459938582725705859473, −5.88766396042881157726493356758, −5.83961248398553963860705521068, −5.54838507311446131940763750617, −5.39836239093837971827370452998, −5.17122501258910507115592066310, −4.82385644311647470037347979052, −4.66189211065899043038366722962, −4.24827113296316988379284838738, −3.90645089385349625246229674718, −3.83164743235685588504676658622, −3.44806444111806495987702984262, −3.38171630551222136045051876527, −3.17608693914723333447999375355, −3.05697446319697971825497965314, −2.63469646881604824933228254168, −2.59826116250856262726436913046, −2.38912370413532841440509019205, −2.32173005205651868303317773901, −1.48722387404717483233693674375, −0.933063568419090143709175370031, −0.932347687983879312543311894919, −0.46642505627728100919598657793,
0.46642505627728100919598657793, 0.932347687983879312543311894919, 0.933063568419090143709175370031, 1.48722387404717483233693674375, 2.32173005205651868303317773901, 2.38912370413532841440509019205, 2.59826116250856262726436913046, 2.63469646881604824933228254168, 3.05697446319697971825497965314, 3.17608693914723333447999375355, 3.38171630551222136045051876527, 3.44806444111806495987702984262, 3.83164743235685588504676658622, 3.90645089385349625246229674718, 4.24827113296316988379284838738, 4.66189211065899043038366722962, 4.82385644311647470037347979052, 5.17122501258910507115592066310, 5.39836239093837971827370452998, 5.54838507311446131940763750617, 5.83961248398553963860705521068, 5.88766396042881157726493356758, 6.29759640459938582725705859473, 6.56508512973980881960217348007, 6.60710582639251470740998581017
