| L(s) = 1 | − 2-s − 5-s − 7-s + 3·9-s + 10-s − 11-s − 13-s + 14-s + 3·17-s − 3·18-s + 22-s + 3·23-s + 26-s − 3·34-s + 35-s − 37-s − 3·45-s − 3·46-s − 47-s + 55-s − 59-s − 61-s − 3·63-s + 65-s − 70-s + 74-s + 77-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s − 7-s + 3·9-s + 10-s − 11-s − 13-s + 14-s + 3·17-s − 3·18-s + 22-s + 3·23-s + 26-s − 3·34-s + 35-s − 37-s − 3·45-s − 3·46-s − 47-s + 55-s − 59-s − 61-s − 3·63-s + 65-s − 70-s + 74-s + 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59776471 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59776471 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2715680575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2715680575\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.35968796110459573323099717942, −9.819933999796422020665851329342, −9.752911443352910155395872880046, −9.547703490173851703373734041208, −9.281225437296678086620470987276, −8.855826839459769908623998270221, −8.483394649709487399319158077524, −8.046240959459131186927590978258, −7.74867286569079367044523927005, −7.33995909206152868120802338742, −7.29799783629349381045532330335, −7.26949194018166010912486430293, −6.54752919336135201359764470213, −6.48572676939444848113121302443, −5.62006424617013831319665945732, −5.42542558682177867693382614112, −4.81242803010083744048334403743, −4.75704942629856563960102810028, −4.36664393270971181929083928963, −3.57359896023615186959525414822, −3.39657580406727644201735611116, −3.19037428923927735858771724883, −2.47577651522300296441081487385, −1.48639734768758000025442494137, −1.10737984500835628978997046673,
1.10737984500835628978997046673, 1.48639734768758000025442494137, 2.47577651522300296441081487385, 3.19037428923927735858771724883, 3.39657580406727644201735611116, 3.57359896023615186959525414822, 4.36664393270971181929083928963, 4.75704942629856563960102810028, 4.81242803010083744048334403743, 5.42542558682177867693382614112, 5.62006424617013831319665945732, 6.48572676939444848113121302443, 6.54752919336135201359764470213, 7.26949194018166010912486430293, 7.29799783629349381045532330335, 7.33995909206152868120802338742, 7.74867286569079367044523927005, 8.046240959459131186927590978258, 8.483394649709487399319158077524, 8.855826839459769908623998270221, 9.281225437296678086620470987276, 9.547703490173851703373734041208, 9.752911443352910155395872880046, 9.819933999796422020665851329342, 10.35968796110459573323099717942
