| L(s) = 1 | − 2-s + 9·3-s + 32·4-s + 34·5-s − 9·6-s − 95·8-s − 34·10-s + 340·11-s + 288·12-s + 908·13-s + 306·15-s + 95·16-s + 798·17-s − 892·19-s + 1.08e3·20-s − 340·22-s + 3.19e3·23-s − 855·24-s + 3.12e3·25-s − 908·26-s − 729·27-s − 1.64e4·29-s − 306·30-s + 2.49e3·31-s − 3.04e3·32-s + 3.06e3·33-s − 798·34-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 0.577·3-s + 4-s + 0.608·5-s − 0.102·6-s − 0.524·8-s − 0.107·10-s + 0.847·11-s + 0.577·12-s + 1.49·13-s + 0.351·15-s + 0.0927·16-s + 0.669·17-s − 0.566·19-s + 0.608·20-s − 0.149·22-s + 1.25·23-s − 0.302·24-s + 25-s − 0.263·26-s − 0.192·27-s − 3.63·29-s − 0.0620·30-s + 0.466·31-s − 0.524·32-s + 0.489·33-s − 0.118·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.135593940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.135593940\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 31 T^{2} + p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 34 T - 1969 T^{2} - 34 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 340 T - 45451 T^{2} - 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 454 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 798 T - 783053 T^{2} - 798 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 892 T - 1680435 T^{2} + 892 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3192 T + 3752521 T^{2} - 3192 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8242 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2496 T - 22399135 T^{2} - 2496 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 9798 T + 26656847 T^{2} + 9798 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 19834 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 17236 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8928 T - 149635823 T^{2} + 8928 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 150 T - 418172993 T^{2} + 150 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 42396 T + 1082496517 T^{2} - 42396 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14758 T - 626797737 T^{2} + 14758 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1676 T - 1347316131 T^{2} - 1676 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14568 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 78378 T + 4070039291 T^{2} + 78378 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2272 T - 3071894415 T^{2} - 2272 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 37764 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 117286 T + 8171946347 T^{2} - 117286 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10002 T + p^{5} T^{2} )^{2} \) |
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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
−12.58533218067522064031375841121, −11.46494446111996955188613008636, −11.46185058634540938349271985442, −11.04683385692800014174231081683, −10.40239734879337093154202076735, −9.759684604346111563874226656577, −9.145710049258621568770070955734, −8.859782024463065321493213918077, −8.429342519160791736982855892701, −7.37102860987291776611890731681, −7.25496542246421484664806699073, −6.41132485697829752175078152583, −5.99840586250065689225929036539, −5.49693741508349489769407815924, −4.45789763170981101844147039679, −3.40301642306557464678778925580, −3.30980044719199808460636837565, −2.08247390718390313233196119436, −1.71804856053153784102608933785, −0.76667496425119150520312119182,
0.76667496425119150520312119182, 1.71804856053153784102608933785, 2.08247390718390313233196119436, 3.30980044719199808460636837565, 3.40301642306557464678778925580, 4.45789763170981101844147039679, 5.49693741508349489769407815924, 5.99840586250065689225929036539, 6.41132485697829752175078152583, 7.25496542246421484664806699073, 7.37102860987291776611890731681, 8.429342519160791736982855892701, 8.859782024463065321493213918077, 9.145710049258621568770070955734, 9.759684604346111563874226656577, 10.40239734879337093154202076735, 11.04683385692800014174231081683, 11.46185058634540938349271985442, 11.46494446111996955188613008636, 12.58533218067522064031375841121
