| L(s) = 1 | − 9·5-s + 5·7-s + 18·9-s + 3·11-s − 15·17-s + 38·19-s − 60·23-s + 25·25-s − 45·35-s − 85·43-s − 162·45-s − 75·47-s + 49·49-s − 27·55-s + 103·61-s + 90·63-s + 25·73-s + 15·77-s + 243·81-s − 180·83-s + 135·85-s − 342·95-s + 54·99-s + 204·101-s + 540·115-s − 75·119-s + 121·121-s + ⋯ |
| L(s) = 1 | − 9/5·5-s + 5/7·7-s + 2·9-s + 3/11·11-s − 0.882·17-s + 2·19-s − 2.60·23-s + 25-s − 9/7·35-s − 1.97·43-s − 3.59·45-s − 1.59·47-s + 49-s − 0.490·55-s + 1.68·61-s + 10/7·63-s + 0.342·73-s + 0.194·77-s + 3·81-s − 2.16·83-s + 1.58·85-s − 3.59·95-s + 6/11·99-s + 2.01·101-s + 4.69·115-s − 0.630·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.421802517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421802517\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T - 24 T^{2} - 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 112 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 85 T + 5376 T^{2} + 85 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T + 3416 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 103 T + 6888 T^{2} - 103 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−9.656161300041797426356212897973, −9.655673269666963791873207697637, −8.602405603414539888927232964383, −8.579596543178915337481251867210, −7.85320533508161270552148029886, −7.82037675583892196228682400844, −7.29001599711649008202969349806, −7.15938970065680280415891923807, −6.46260897176344204746417173242, −6.19187828276540075727027518694, −5.16723219614876756657440988069, −5.10892463903404308547963880694, −4.32363852995342092160394397940, −4.20389069497851144119118518502, −3.62831969914893079497934577170, −3.49134520444998288406440769099, −2.40414402109962889480599714349, −1.71954298559270567831099226396, −1.30036670126043883232048030226, −0.37696500066613395303721267576,
0.37696500066613395303721267576, 1.30036670126043883232048030226, 1.71954298559270567831099226396, 2.40414402109962889480599714349, 3.49134520444998288406440769099, 3.62831969914893079497934577170, 4.20389069497851144119118518502, 4.32363852995342092160394397940, 5.10892463903404308547963880694, 5.16723219614876756657440988069, 6.19187828276540075727027518694, 6.46260897176344204746417173242, 7.15938970065680280415891923807, 7.29001599711649008202969349806, 7.82037675583892196228682400844, 7.85320533508161270552148029886, 8.579596543178915337481251867210, 8.602405603414539888927232964383, 9.655673269666963791873207697637, 9.656161300041797426356212897973
