| L(s) = 1 | − 4·2-s + 9·3-s − 6·5-s − 36·6-s + 64·8-s + 24·10-s + 666·11-s + 1.11e3·13-s − 54·15-s − 256·16-s − 1.74e3·17-s + 1.15e3·19-s − 2.66e3·22-s + 3.46e3·23-s + 576·24-s + 3.12e3·25-s − 4.47e3·26-s − 729·27-s + 6.74e3·29-s + 216·30-s + 6.29e3·31-s + 5.99e3·33-s + 6.96e3·34-s − 3.13e3·37-s − 4.62e3·38-s + 1.00e4·39-s − 384·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.107·5-s − 0.408·6-s + 0.353·8-s + 0.0758·10-s + 1.65·11-s + 1.83·13-s − 0.0619·15-s − 1/4·16-s − 1.46·17-s + 0.735·19-s − 1.17·22-s + 1.36·23-s + 0.204·24-s + 25-s − 1.29·26-s − 0.192·27-s + 1.48·29-s + 0.0438·30-s + 1.17·31-s + 0.958·33-s + 1.03·34-s − 0.375·37-s − 0.519·38-s + 1.05·39-s − 0.0379·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.733317693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.733317693\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T - 3089 T^{2} + 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 666 T + 282505 T^{2} - 666 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 43 p T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1740 T + 1607743 T^{2} + 1740 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 1157 T - 1137450 T^{2} - 1157 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3468 T + 5590681 T^{2} - 3468 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3372 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 203 p T + 11418 p^{2} T^{2} - 203 p^{6} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3131 T - 59540796 T^{2} + 3131 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4866 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11407 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2310 T - 224008907 T^{2} - 2310 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 28296 T + 382468123 T^{2} - 28296 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20544 T - 292868363 T^{2} - 20544 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4630 T - 823159401 T^{2} + 4630 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 18745 T - 998750082 T^{2} - 18745 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 38226 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 70589 T + 2909735328 T^{2} - 70589 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 62293 T + 803361450 T^{2} - 62293 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 79818 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18120 T - 5255725049 T^{2} + 18120 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 124754 T + p^{5} T^{2} )^{2} \) |
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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
−11.03567748981070995046271770462, −10.85368577207167651578553297938, −10.08322346189933902031858603593, −9.621875835566487876493018268320, −9.155231460146695749733615012799, −8.616489108274838051287344050981, −8.539866121027255912603647154679, −8.236451994783072162007565001063, −7.16229498861651554853591421634, −6.70050611261365736480099378246, −6.66340240983807495978096858029, −5.82678148160099584910215572988, −5.06349750084131179402539717566, −4.31195164346514827120201283176, −4.02709157117137369075997123664, −3.18479114723959621418810255129, −2.75980179333151237766430056604, −1.66136784998079231227713008756, −1.11591644617717784554228576472, −0.69570274335452160640611942480,
0.69570274335452160640611942480, 1.11591644617717784554228576472, 1.66136784998079231227713008756, 2.75980179333151237766430056604, 3.18479114723959621418810255129, 4.02709157117137369075997123664, 4.31195164346514827120201283176, 5.06349750084131179402539717566, 5.82678148160099584910215572988, 6.66340240983807495978096858029, 6.70050611261365736480099378246, 7.16229498861651554853591421634, 8.236451994783072162007565001063, 8.539866121027255912603647154679, 8.616489108274838051287344050981, 9.155231460146695749733615012799, 9.621875835566487876493018268320, 10.08322346189933902031858603593, 10.85368577207167651578553297938, 11.03567748981070995046271770462
