| L(s) = 1 | − 6·3-s − 10·5-s − 2·7-s + 27·9-s + 22·11-s − 14·13-s + 60·15-s − 80·17-s + 60·19-s + 12·21-s − 202·23-s − 78·25-s − 108·27-s + 336·29-s + 128·31-s − 132·33-s + 20·35-s − 188·37-s + 84·39-s − 132·41-s − 480·43-s − 270·45-s − 390·47-s + 190·49-s + 480·51-s − 610·53-s − 220·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.107·7-s + 9-s + 0.603·11-s − 0.298·13-s + 1.03·15-s − 1.14·17-s + 0.724·19-s + 0.124·21-s − 1.83·23-s − 0.623·25-s − 0.769·27-s + 2.15·29-s + 0.741·31-s − 0.696·33-s + 0.0965·35-s − 0.835·37-s + 0.344·39-s − 0.502·41-s − 1.70·43-s − 0.894·45-s − 1.21·47-s + 0.553·49-s + 1.31·51-s − 1.58·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 p T + 178 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 186 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14 T - 310 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 80 T + 11038 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 60 T + 4918 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 202 T + 32110 T^{2} + 202 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 336 T + 73510 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 128 T + 38846 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 188 T + 10814 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 86326 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 480 T + 210406 T^{2} + 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 390 T + 244798 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 610 T + 271954 T^{2} + 610 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 372 T + 413926 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1050 T + 724834 T^{2} + 1050 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 408 T + 618310 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 942 T + 915838 T^{2} - 942 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 740 T + 517622 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1646 T + 1592694 T^{2} + 1646 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 352 T + 912262 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2036 T + 2421430 T^{2} - 2036 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 336 T + 1378270 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.382237213256228231013984527590, −8.182707130279089129297955822091, −7.70008327213610127982627345895, −7.47414219236158627386096786820, −6.76970340206890302487017353251, −6.56056196783373234864749952384, −6.22134116108076510612019552771, −6.03205930790491313815922182853, −5.14484362643326301048389796722, −4.98060490597833020508790923643, −4.42263863939471483997892550557, −4.33349005229228940253186169933, −3.54284346884045824917170717624, −3.35203080412582317722069261714, −2.63587068068512535629823100462, −1.83844844051721558291105116800, −1.55788018956457229539610301267, −0.76259521051097788393769538872, 0, 0,
0.76259521051097788393769538872, 1.55788018956457229539610301267, 1.83844844051721558291105116800, 2.63587068068512535629823100462, 3.35203080412582317722069261714, 3.54284346884045824917170717624, 4.33349005229228940253186169933, 4.42263863939471483997892550557, 4.98060490597833020508790923643, 5.14484362643326301048389796722, 6.03205930790491313815922182853, 6.22134116108076510612019552771, 6.56056196783373234864749952384, 6.76970340206890302487017353251, 7.47414219236158627386096786820, 7.70008327213610127982627345895, 8.182707130279089129297955822091, 8.382237213256228231013984527590
