| L(s) = 1 | − 2-s + 4-s + 8·5-s − 8-s − 6·9-s − 8·10-s − 4·13-s + 16-s − 4·17-s + 6·18-s + 8·20-s + 38·25-s + 4·26-s + 4·29-s − 32-s + 4·34-s − 6·36-s − 8·37-s − 8·40-s + 12·41-s − 48·45-s + 2·49-s − 38·50-s − 4·52-s − 8·53-s − 4·58-s − 16·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 3.57·5-s − 0.353·8-s − 2·9-s − 2.52·10-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 1.41·18-s + 1.78·20-s + 38/5·25-s + 0.784·26-s + 0.742·29-s − 0.176·32-s + 0.685·34-s − 36-s − 1.31·37-s − 1.26·40-s + 1.87·41-s − 7.15·45-s + 2/7·49-s − 5.37·50-s − 0.554·52-s − 1.09·53-s − 0.525·58-s − 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.201795812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201795812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−10.68153496695016269554639136746, −10.54062617762270704890133609515, −9.594811427439617650906242640583, −9.573778240853111358913501663130, −8.971935452602447588346290189052, −8.689636804269542263141419996184, −7.79508532540708076585721889920, −6.69441913641732873591204403077, −6.43078276958769513552458688049, −5.84432417959727908710252168629, −5.40962343507111122438401931438, −4.88798562712811078749313553550, −2.71817369175489666462579434927, −2.63232737474552247848840602025, −1.74138515374411887221562842274,
1.74138515374411887221562842274, 2.63232737474552247848840602025, 2.71817369175489666462579434927, 4.88798562712811078749313553550, 5.40962343507111122438401931438, 5.84432417959727908710252168629, 6.43078276958769513552458688049, 6.69441913641732873591204403077, 7.79508532540708076585721889920, 8.689636804269542263141419996184, 8.971935452602447588346290189052, 9.573778240853111358913501663130, 9.594811427439617650906242640583, 10.54062617762270704890133609515, 10.68153496695016269554639136746
