| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 2·6-s − 2·7-s + 3·8-s + 3·9-s + 2·11-s + 4·12-s + 2·13-s + 2·14-s + 16-s − 6·17-s − 3·18-s + 4·21-s − 2·22-s + 2·23-s − 6·24-s − 2·26-s − 4·27-s + 4·28-s + 10·29-s − 6·31-s − 2·32-s − 4·33-s + 6·34-s − 6·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 9-s + 0.603·11-s + 1.15·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.872·21-s − 0.426·22-s + 0.417·23-s − 1.22·24-s − 0.392·26-s − 0.769·27-s + 0.755·28-s + 1.85·29-s − 1.07·31-s − 0.353·32-s − 0.696·33-s + 1.02·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33350625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33350625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5561644268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5561644268\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 24 T + 273 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 207 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 242 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.274275376760460704146746634063, −8.208520660486485694787371629714, −7.45552283948838077710272353991, −7.21593339437810307273754383012, −6.85663049370700925035941373323, −6.50404685730716332838904591580, −6.12206141581698848591972037019, −5.97195037148174805710929971665, −5.35422831630737534132414259856, −5.16247113113174703653159966572, −4.48211294095250253063893064027, −4.29166831624316656138408870930, −4.13605423404094361011755694007, −3.67682321554640306035754501984, −2.87033988836941557577167206047, −2.65668727198678965554497583395, −1.93329324593234579873764040439, −1.18871771788384588926628286648, −0.861402166253848872451509356303, −0.35356167827866478505652865529,
0.35356167827866478505652865529, 0.861402166253848872451509356303, 1.18871771788384588926628286648, 1.93329324593234579873764040439, 2.65668727198678965554497583395, 2.87033988836941557577167206047, 3.67682321554640306035754501984, 4.13605423404094361011755694007, 4.29166831624316656138408870930, 4.48211294095250253063893064027, 5.16247113113174703653159966572, 5.35422831630737534132414259856, 5.97195037148174805710929971665, 6.12206141581698848591972037019, 6.50404685730716332838904591580, 6.85663049370700925035941373323, 7.21593339437810307273754383012, 7.45552283948838077710272353991, 8.208520660486485694787371629714, 8.274275376760460704146746634063
