| L(s) = 1 | − 2-s + 2·3-s − 4-s − 3·5-s − 2·6-s + 3·7-s + 3·8-s + 3·9-s + 3·10-s − 2·12-s + 4·13-s − 3·14-s − 6·15-s − 16-s − 4·17-s − 3·18-s + 3·20-s + 6·21-s + 6·24-s + 2·25-s − 4·26-s + 4·27-s − 3·28-s + 8·29-s + 6·30-s + 4·31-s − 5·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.34·5-s − 0.816·6-s + 1.13·7-s + 1.06·8-s + 9-s + 0.948·10-s − 0.577·12-s + 1.10·13-s − 0.801·14-s − 1.54·15-s − 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.670·20-s + 1.30·21-s + 1.22·24-s + 2/5·25-s − 0.784·26-s + 0.769·27-s − 0.566·28-s + 1.48·29-s + 1.09·30-s + 0.718·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8340510846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8340510846\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−16.5551253122, −15.8734598001, −15.5668057746, −15.3135190512, −14.5135323404, −14.1389861990, −13.7536561447, −13.1937925053, −12.6955921041, −11.8071100067, −11.5559508175, −10.7951208381, −10.4382147099, −9.72466121073, −8.82668426743, −8.67236519668, −8.23439647711, −7.75866687526, −7.25047783803, −6.39854886740, −5.11098268656, −4.23445324460, −4.13559084051, −2.94053502253, −1.47970352315,
1.47970352315, 2.94053502253, 4.13559084051, 4.23445324460, 5.11098268656, 6.39854886740, 7.25047783803, 7.75866687526, 8.23439647711, 8.67236519668, 8.82668426743, 9.72466121073, 10.4382147099, 10.7951208381, 11.5559508175, 11.8071100067, 12.6955921041, 13.1937925053, 13.7536561447, 14.1389861990, 14.5135323404, 15.3135190512, 15.5668057746, 15.8734598001, 16.5551253122
