| 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
| \(L(3)\) |
\(\approx\) |
\(0.1258927105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1258927105\) |
| \(L(\frac{7}{2})\) |
|
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
−12.54321727882313697855467309440, −11.71426069185043276902069386303, −11.47433714165559871693328137385, −11.00168931478824423519784279495, −10.64530021488145323892153373047, −9.626691942940905043951850174883, −9.548853008191748536800693788864, −8.722738312335438389974241508886, −8.275079812084057674235200964286, −7.19629727661128669694281016612, −7.03429203284902412971283531114, −6.86287296352405389711436577892, −5.90440081564089558128575018383, −5.00337794278833868111779082772, −4.97580495163268527437942507653, −3.49124243158454700518046662289, −3.36170736378964369911049356106, −2.07938356322734420877207094939, −1.61001454607709821291771954059, −0.12097618969151693032224129828,
0.12097618969151693032224129828, 1.61001454607709821291771954059, 2.07938356322734420877207094939, 3.36170736378964369911049356106, 3.49124243158454700518046662289, 4.97580495163268527437942507653, 5.00337794278833868111779082772, 5.90440081564089558128575018383, 6.86287296352405389711436577892, 7.03429203284902412971283531114, 7.19629727661128669694281016612, 8.275079812084057674235200964286, 8.722738312335438389974241508886, 9.548853008191748536800693788864, 9.626691942940905043951850174883, 10.64530021488145323892153373047, 11.00168931478824423519784279495, 11.47433714165559871693328137385, 11.71426069185043276902069386303, 12.54321727882313697855467309440
