| L(s) = 1 | + 2·3-s + 24·7-s + 27·9-s − 24·11-s + 91·13-s − 117·17-s + 198·19-s + 48·21-s − 78·23-s + 247·25-s + 154·27-s + 141·29-s − 48·33-s − 249·37-s + 182·39-s + 471·41-s + 104·43-s + 41·49-s − 234·51-s + 186·53-s + 396·57-s + 492·59-s − 145·61-s + 648·63-s + 1.36e3·67-s − 156·69-s − 1.83e3·71-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.29·7-s + 9-s − 0.657·11-s + 1.94·13-s − 1.66·17-s + 2.39·19-s + 0.498·21-s − 0.707·23-s + 1.97·25-s + 1.09·27-s + 0.902·29-s − 0.253·33-s − 1.10·37-s + 0.747·39-s + 1.79·41-s + 0.368·43-s + 0.119·49-s − 0.642·51-s + 0.482·53-s + 0.920·57-s + 1.08·59-s − 0.304·61-s + 1.29·63-s + 2.48·67-s − 0.272·69-s − 3.05·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.337233776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.337233776\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 247 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 535 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 24 T + 1523 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 117 T + 8776 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 198 T + 19927 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 78 T - 6083 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 141 T - 4508 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 249 T + 71320 T^{2} + 249 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 471 T + 142868 T^{2} - 471 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 104 T - 68691 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 116818 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 492 T + 286067 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1362 T + 919111 T^{2} - 1362 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1830 T + 1474211 T^{2} + 1830 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 567359 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1276 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 519766 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1692 T + 1659257 T^{2} + 1692 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 348 T + 953041 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.14674524305745409670587747741, −11.56885243740790785001834536092, −11.09267344104038232102105219269, −10.89733847083912375357617703906, −10.25109597878612378579809896786, −9.835781014522031616098884338933, −8.934129358721195011190726339742, −8.727149893366397299792231971338, −8.283761099719966662700360282901, −7.72567831698769910932769734207, −6.92495851557559707457824789907, −6.89956670413125727925317936518, −5.77872364940037632261564910583, −5.35594562269790954217406331365, −4.43817892699905125639082933181, −4.34095374656694777953470558090, −3.24767138576359516999663601721, −2.60138688882169316357273968957, −1.45029455959214390114118125828, −1.04631267028510669576583039583,
1.04631267028510669576583039583, 1.45029455959214390114118125828, 2.60138688882169316357273968957, 3.24767138576359516999663601721, 4.34095374656694777953470558090, 4.43817892699905125639082933181, 5.35594562269790954217406331365, 5.77872364940037632261564910583, 6.89956670413125727925317936518, 6.92495851557559707457824789907, 7.72567831698769910932769734207, 8.283761099719966662700360282901, 8.727149893366397299792231971338, 8.934129358721195011190726339742, 9.835781014522031616098884338933, 10.25109597878612378579809896786, 10.89733847083912375357617703906, 11.09267344104038232102105219269, 11.56885243740790785001834536092, 12.14674524305745409670587747741
