| L(s) = 1 | − 9·5-s − 31·7-s + 15·11-s + 37·13-s + 84·17-s + 56·19-s − 195·23-s + 125·25-s + 111·29-s − 205·31-s + 279·35-s − 332·37-s − 261·41-s − 43·43-s − 177·47-s + 343·49-s − 228·53-s − 135·55-s − 159·59-s − 191·61-s − 333·65-s − 421·67-s + 312·71-s + 364·73-s − 465·77-s + 1.13e3·79-s + 1.08e3·83-s + ⋯ |
| L(s) = 1 | − 0.804·5-s − 1.67·7-s + 0.411·11-s + 0.789·13-s + 1.19·17-s + 0.676·19-s − 1.76·23-s + 25-s + 0.710·29-s − 1.18·31-s + 1.34·35-s − 1.47·37-s − 0.994·41-s − 0.152·43-s − 0.549·47-s + 49-s − 0.590·53-s − 0.330·55-s − 0.350·59-s − 0.400·61-s − 0.635·65-s − 0.767·67-s + 0.521·71-s + 0.583·73-s − 0.688·77-s + 1.61·79-s + 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8863012641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8863012641\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 9 T - 44 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 31 T + 618 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 37 T - 828 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 195 T + 25858 T^{2} + 195 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 111 T - 12068 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 205 T + 12234 T^{2} + 205 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 261 T - 800 T^{2} + 261 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + p T - 42 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 177 T - 72494 T^{2} + 177 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 159 T - 180098 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 191 T - 190500 T^{2} + 191 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 421 T - 123522 T^{2} + 421 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 156 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1133 T + 790650 T^{2} - 1133 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1083 T + 601102 T^{2} - 1083 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1050 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 901 T - 100872 T^{2} - 901 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.77917725759148092271318369268, −10.40894919483553110190630612972, −10.23148323017230714257345455513, −9.410260275127040418797247643904, −9.343681836236453533628441478390, −8.774372866397727631277820961578, −7.969441525830167702290528977839, −7.965799518611042591148609586694, −7.24356115688758323629364233179, −6.55978718274516047111938178397, −6.48104280476927233795848594527, −5.83381891554021431946591178676, −5.24115256974848884317836032062, −4.63672822057916849006697735645, −3.69613215484793512993523963489, −3.42685737800402894206051423204, −3.31349511894806453452290979204, −2.13365243459226088686900512136, −1.23253564219819021721158489995, −0.33556561062097868471125577813,
0.33556561062097868471125577813, 1.23253564219819021721158489995, 2.13365243459226088686900512136, 3.31349511894806453452290979204, 3.42685737800402894206051423204, 3.69613215484793512993523963489, 4.63672822057916849006697735645, 5.24115256974848884317836032062, 5.83381891554021431946591178676, 6.48104280476927233795848594527, 6.55978718274516047111938178397, 7.24356115688758323629364233179, 7.965799518611042591148609586694, 7.969441525830167702290528977839, 8.774372866397727631277820961578, 9.343681836236453533628441478390, 9.410260275127040418797247643904, 10.23148323017230714257345455513, 10.40894919483553110190630612972, 10.77917725759148092271318369268
