| L(s) = 1 | − 2·5-s − 4·7-s + 4·11-s + 2·13-s − 12·17-s + 8·19-s + 5·25-s − 2·29-s + 4·31-s + 8·35-s − 4·37-s − 2·41-s + 4·43-s + 8·47-s + 7·49-s + 20·53-s − 8·55-s − 4·59-s − 6·61-s − 4·65-s + 4·67-s + 32·71-s − 12·73-s − 16·77-s + 4·79-s + 12·83-s + 24·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s − 2.91·17-s + 1.83·19-s + 25-s − 0.371·29-s + 0.718·31-s + 1.35·35-s − 0.657·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 49-s + 2.74·53-s − 1.07·55-s − 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 3.79·71-s − 1.40·73-s − 1.82·77-s + 0.450·79-s + 1.31·83-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.781610527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781610527\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.181260432532732230814547556802, −8.863815422335461484140769861392, −8.439695542572196120843752660478, −7.949099171204804054231274746627, −7.30867426172839880924360364016, −7.16933318112677951996644411090, −6.81172105259731850726190815494, −6.41338395776182372350631342531, −6.18179663651632854151098871938, −5.72727553275363619569350053831, −4.92414258079304070910957523904, −4.83945372803317887353957767573, −4.02102174716011838081971847687, −3.93471193375828351093264199323, −3.44563488727206024753287154811, −3.10835418919690462636411207882, −2.36064359754536526197929476651, −2.02995788738158951523044803601, −0.882638062411890367161897780691, −0.59340222245759086647494295014,
0.59340222245759086647494295014, 0.882638062411890367161897780691, 2.02995788738158951523044803601, 2.36064359754536526197929476651, 3.10835418919690462636411207882, 3.44563488727206024753287154811, 3.93471193375828351093264199323, 4.02102174716011838081971847687, 4.83945372803317887353957767573, 4.92414258079304070910957523904, 5.72727553275363619569350053831, 6.18179663651632854151098871938, 6.41338395776182372350631342531, 6.81172105259731850726190815494, 7.16933318112677951996644411090, 7.30867426172839880924360364016, 7.949099171204804054231274746627, 8.439695542572196120843752660478, 8.863815422335461484140769861392, 9.181260432532732230814547556802
