| L(s) = 1 | − 2·4-s − 2·7-s − 16·13-s + 4·16-s + 2·19-s + 4·25-s + 4·28-s + 14·31-s − 16·37-s − 4·43-s − 11·49-s + 32·52-s − 10·61-s − 16·64-s − 4·67-s + 2·73-s − 4·76-s + 8·79-s + 32·91-s − 4·97-s − 8·100-s − 4·103-s + 2·109-s − 8·112-s − 2·121-s − 28·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 4.43·13-s + 16-s + 0.458·19-s + 4/5·25-s + 0.755·28-s + 2.51·31-s − 2.63·37-s − 0.609·43-s − 1.57·49-s + 4.43·52-s − 1.28·61-s − 2·64-s − 0.488·67-s + 0.234·73-s − 0.458·76-s + 0.900·79-s + 3.35·91-s − 0.406·97-s − 4/5·100-s − 0.394·103-s + 0.191·109-s − 0.755·112-s − 0.181·121-s − 2.51·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3770153081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3770153081\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 88 T^{2} + 5535 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 100 T^{2} + 7191 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 22 T^{2} - 2997 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
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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
−9.386107836959756905613687760093, −8.898919307860631113655323304460, −8.741912310300545181441345269219, −8.496341649764835168194205485844, −8.007457901723296254136879527838, −7.994620343981560846480658472270, −7.38437923010358338865086347610, −7.35260581050469289555805905078, −7.20533030033003340705467831477, −6.68920028234694589583250605058, −6.60590713056930433224715436838, −6.10988063794248592995902091945, −5.84067678402851654791515757030, −5.25733256871539093432632858317, −5.04974381995710404371216687153, −4.87215554695444942860836066911, −4.73718544340895095236392094312, −4.40056290502220437786994293229, −3.90766409721401139967098081221, −3.21494881838272153135875681200, −3.04210186654321559409953884970, −2.85818645579476803603353314665, −2.22569423766529871921410543048, −1.67767735000749869672040150176, −0.39402985270323709124190149943,
0.39402985270323709124190149943, 1.67767735000749869672040150176, 2.22569423766529871921410543048, 2.85818645579476803603353314665, 3.04210186654321559409953884970, 3.21494881838272153135875681200, 3.90766409721401139967098081221, 4.40056290502220437786994293229, 4.73718544340895095236392094312, 4.87215554695444942860836066911, 5.04974381995710404371216687153, 5.25733256871539093432632858317, 5.84067678402851654791515757030, 6.10988063794248592995902091945, 6.60590713056930433224715436838, 6.68920028234694589583250605058, 7.20533030033003340705467831477, 7.35260581050469289555805905078, 7.38437923010358338865086347610, 7.994620343981560846480658472270, 8.007457901723296254136879527838, 8.496341649764835168194205485844, 8.741912310300545181441345269219, 8.898919307860631113655323304460, 9.386107836959756905613687760093
