| L(s) = 1 | + 3·3-s − 7·5-s − 35·7-s − 7·11-s − 104·13-s − 21·15-s − 72·17-s − 20·19-s − 105·21-s + 48·23-s + 125·25-s − 27·27-s − 486·29-s − 95·31-s − 21·33-s + 245·35-s − 352·37-s − 312·39-s − 592·41-s + 316·43-s + 142·47-s + 882·49-s − 216·51-s + 375·53-s + 49·55-s − 60·57-s − 279·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.626·5-s − 1.88·7-s − 0.191·11-s − 2.21·13-s − 0.361·15-s − 1.02·17-s − 0.241·19-s − 1.09·21-s + 0.435·23-s + 25-s − 0.192·27-s − 3.11·29-s − 0.550·31-s − 0.110·33-s + 1.18·35-s − 1.56·37-s − 1.28·39-s − 2.25·41-s + 1.12·43-s + 0.440·47-s + 18/7·49-s − 0.593·51-s + 0.971·53-s + 0.120·55-s − 0.139·57-s − 0.615·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04757057571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04757057571\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T - 1282 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 72 T + 271 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T - 6459 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 48 T - 9863 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 243 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 95 T - 20766 T^{2} + 95 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 352 T + 73251 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 296 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 158 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 142 T - 83659 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 375 T - 8252 T^{2} - 375 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 279 T - 127538 T^{2} + 279 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 246 T - 166465 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 730 T + 232137 T^{2} - 730 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 338 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 542 T - 95253 T^{2} - 542 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 305 T - 400014 T^{2} - 305 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1123 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 426 T - 523493 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 369 T + p^{3} T^{2} )^{2} \) |
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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
−12.95574866949415931667518983934, −12.23287969488609163581650358251, −11.78780327194156947775232896037, −10.80704302418185671473093829527, −10.74972295750534037211343528596, −9.852954440113651743003568241464, −9.548466743749313469442305492463, −9.021143167181959073609136871328, −8.771601431450838547590827257493, −7.58124339442813279129993966042, −7.55251927847249797867569448422, −6.73233831751785511956851851938, −6.58005714731960238806626554003, −5.24395895617847098470568138322, −5.17677307119239961691101714524, −3.77117201079586805514176180305, −3.70420376182731715479725051019, −2.66662254961118633936483250760, −2.17095810359329923443057299926, −0.089907210204256695221940630583,
0.089907210204256695221940630583, 2.17095810359329923443057299926, 2.66662254961118633936483250760, 3.70420376182731715479725051019, 3.77117201079586805514176180305, 5.17677307119239961691101714524, 5.24395895617847098470568138322, 6.58005714731960238806626554003, 6.73233831751785511956851851938, 7.55251927847249797867569448422, 7.58124339442813279129993966042, 8.771601431450838547590827257493, 9.021143167181959073609136871328, 9.548466743749313469442305492463, 9.852954440113651743003568241464, 10.74972295750534037211343528596, 10.80704302418185671473093829527, 11.78780327194156947775232896037, 12.23287969488609163581650358251, 12.95574866949415931667518983934
