| L(s) = 1 | − 2-s − 3·7-s − 9-s + 11-s + 3·14-s + 18-s − 22-s − 25-s − 3·31-s + 32-s + 3·37-s − 2·41-s − 8·43-s + 2·47-s + 6·49-s + 50-s − 3·59-s − 2·61-s + 3·62-s + 3·63-s − 64-s − 73-s − 3·74-s − 3·77-s + 2·82-s + 2·83-s + 8·86-s + ⋯ |
| L(s) = 1 | − 2-s − 3·7-s − 9-s + 11-s + 3·14-s + 18-s − 22-s − 25-s − 3·31-s + 32-s + 3·37-s − 2·41-s − 8·43-s + 2·47-s + 6·49-s + 50-s − 3·59-s − 2·61-s + 3·62-s + 3·63-s − 64-s − 73-s − 3·74-s − 3·77-s + 2·82-s + 2·83-s + 8·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{4} \cdot 73^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{4} \cdot 73^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1180985297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1180985297\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{8} \) |
| 47 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−6.23231172520443818949425778563, −6.22983070906969546138606153782, −6.18036700290263034139460513022, −5.73535219192455668345377823805, −5.66180388022044742375924165671, −5.45455929461245158568315859091, −5.16314355439225247286644303303, −4.89759338108894876186291169540, −4.82210720344337623272482569803, −4.35400717735483284649045993171, −4.16328003738146002916565145853, −4.10186838518736965565969966210, −3.57636566454529048207354357821, −3.55949548297736436238917983939, −3.37145927347232463420534637644, −3.09443894863411526352505514518, −3.00331304236830839603875736818, −2.93832899142113477209500750151, −2.41103366202680762136361912763, −2.16677639825147288803562924617, −1.68705398750517909582376185095, −1.52187505668167729706237672973, −1.48894370350928380484818357306, −0.46221256693642683581030494293, −0.31802878561419259467559198191,
0.31802878561419259467559198191, 0.46221256693642683581030494293, 1.48894370350928380484818357306, 1.52187505668167729706237672973, 1.68705398750517909582376185095, 2.16677639825147288803562924617, 2.41103366202680762136361912763, 2.93832899142113477209500750151, 3.00331304236830839603875736818, 3.09443894863411526352505514518, 3.37145927347232463420534637644, 3.55949548297736436238917983939, 3.57636566454529048207354357821, 4.10186838518736965565969966210, 4.16328003738146002916565145853, 4.35400717735483284649045993171, 4.82210720344337623272482569803, 4.89759338108894876186291169540, 5.16314355439225247286644303303, 5.45455929461245158568315859091, 5.66180388022044742375924165671, 5.73535219192455668345377823805, 6.18036700290263034139460513022, 6.22983070906969546138606153782, 6.23231172520443818949425778563
