| L(s) = 1 | − 2·3-s − 2·5-s − 20·7-s − 3·9-s − 22·11-s − 80·13-s + 4·15-s − 124·17-s + 72·19-s + 40·21-s + 98·23-s − 55·25-s − 34·27-s − 144·29-s + 34·31-s + 44·33-s + 40·35-s − 54·37-s + 160·39-s + 536·41-s − 60·43-s + 6·45-s + 272·47-s − 338·49-s + 248·51-s + 492·53-s + 44·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.178·5-s − 1.07·7-s − 1/9·9-s − 0.603·11-s − 1.70·13-s + 0.0688·15-s − 1.76·17-s + 0.869·19-s + 0.415·21-s + 0.888·23-s − 0.439·25-s − 0.242·27-s − 0.922·29-s + 0.196·31-s + 0.232·33-s + 0.193·35-s − 0.239·37-s + 0.656·39-s + 2.04·41-s − 0.212·43-s + 0.0198·45-s + 0.844·47-s − 0.985·49-s + 0.680·51-s + 1.27·53-s + 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 59 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13238 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4214 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 98 T + 22847 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 34 T + 57519 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 54 T + 101843 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 60 T + 159146 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 492 T + 348862 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 634 T + 458975 T^{2} - 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 678 T + 813415 T^{2} - 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 400 T + 160962 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 p T - 279966 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 468 T + 1155130 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.820436952868513215900640252292, −9.543707805821427883112768409871, −8.993535647247031443492625662544, −8.728770485354961352950967708057, −7.973653106166331592383166156373, −7.52573081541208554762112718680, −7.07385886845256128312947304281, −6.98104702000654004914294546040, −6.13366875054024337890250059309, −5.93505387116470112828695600926, −5.06843747953136720392544107950, −5.06349433239766287488666978182, −4.26384815178024334441085208081, −3.79531986110723355823768750031, −3.06220544618917152689940345972, −2.48844991868379492683675693891, −2.20353949561051102218102136109, −0.993080370715134216313539176988, 0, 0,
0.993080370715134216313539176988, 2.20353949561051102218102136109, 2.48844991868379492683675693891, 3.06220544618917152689940345972, 3.79531986110723355823768750031, 4.26384815178024334441085208081, 5.06349433239766287488666978182, 5.06843747953136720392544107950, 5.93505387116470112828695600926, 6.13366875054024337890250059309, 6.98104702000654004914294546040, 7.07385886845256128312947304281, 7.52573081541208554762112718680, 7.973653106166331592383166156373, 8.728770485354961352950967708057, 8.993535647247031443492625662544, 9.543707805821427883112768409871, 9.820436952868513215900640252292
