| L(s) = 1 | − 2·2-s − 11·4-s − 20·5-s + 36·8-s + 40·10-s + 20·11-s + 104·13-s + 61·16-s − 116·17-s + 192·19-s + 220·20-s − 40·22-s − 28·23-s + 148·25-s − 208·26-s − 296·29-s − 104·31-s − 358·32-s + 232·34-s − 248·37-s − 384·38-s − 720·40-s − 20·41-s − 720·43-s − 220·44-s + 56·46-s + 96·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.37·4-s − 1.78·5-s + 1.59·8-s + 1.26·10-s + 0.548·11-s + 2.21·13-s + 0.953·16-s − 1.65·17-s + 2.31·19-s + 2.45·20-s − 0.387·22-s − 0.253·23-s + 1.18·25-s − 1.56·26-s − 1.89·29-s − 0.602·31-s − 1.97·32-s + 1.17·34-s − 1.10·37-s − 1.63·38-s − 2.84·40-s − 0.0761·41-s − 2.55·43-s − 0.753·44-s + 0.179·46-s + 0.297·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 15 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 p T + 252 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 20 T + 1610 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 p T + 5848 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 116 T + 9140 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 192 T + 21966 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 28 T + 22962 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 296 T + 62994 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 104 T + 57286 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 248 T + 112074 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 20 T + 42020 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 720 T + 268614 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 96 T + 84950 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 268 T + 56510 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 616 T + 403470 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 454008 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 144 T + 57558 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 988 T + 852210 T^{2} + 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 104 T + 459136 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 944 T + 1097470 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1016 T + 1388838 T^{2} + 1016 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 388 T + 1217732 T^{2} - 388 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 488 T + 1167280 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.33419811440724983581609669394, −10.05657068243529147717160229682, −9.245276754124986470721385124011, −9.088274030991434728730107760875, −8.687597444370195160304228180508, −8.357014619601616012426625719252, −7.82400630255851885196546771068, −7.51279839970678956632749971633, −6.93810883557655921831242610460, −6.39273802010921929667185561391, −5.37504269197936174321461012309, −5.33814560814580709424506664377, −4.25862134111062709173977883295, −4.14569041347904889351715367624, −3.47535521764679561694509680839, −3.37722073266509372629783289847, −1.60954362499798667457814268582, −1.16261129241548893594957761267, 0, 0,
1.16261129241548893594957761267, 1.60954362499798667457814268582, 3.37722073266509372629783289847, 3.47535521764679561694509680839, 4.14569041347904889351715367624, 4.25862134111062709173977883295, 5.33814560814580709424506664377, 5.37504269197936174321461012309, 6.39273802010921929667185561391, 6.93810883557655921831242610460, 7.51279839970678956632749971633, 7.82400630255851885196546771068, 8.357014619601616012426625719252, 8.687597444370195160304228180508, 9.088274030991434728730107760875, 9.245276754124986470721385124011, 10.05657068243529147717160229682, 10.33419811440724983581609669394
