| L(s) = 1 | − 4-s − 4·5-s − 2·7-s + 2·8-s + 9-s + 4·13-s + 16-s − 4·17-s + 4·20-s + 8·23-s + 2·25-s + 2·28-s − 4·29-s − 4·32-s + 8·35-s − 36-s − 4·37-s − 8·40-s − 4·41-s − 8·43-s − 4·45-s + 3·49-s − 4·52-s + 12·53-s − 4·56-s + 16·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s − 0.755·7-s + 0.707·8-s + 1/3·9-s + 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.894·20-s + 1.66·23-s + 2/5·25-s + 0.377·28-s − 0.742·29-s − 0.707·32-s + 1.35·35-s − 1/6·36-s − 0.657·37-s − 1.26·40-s − 0.624·41-s − 1.21·43-s − 0.596·45-s + 3/7·49-s − 0.554·52-s + 1.64·53-s − 0.534·56-s + 2.08·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3919569721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3919569721\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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
−19.8970146357, −19.6678488176, −19.2406986299, −18.4485210975, −18.4192282680, −17.1598241820, −16.8741179353, −15.8734598001, −15.8196150869, −15.2472562838, −14.5135323404, −13.4821859369, −13.1937925053, −12.6521323881, −11.5559508175, −11.3731892768, −10.5195197796, −9.72466121073, −8.67236519668, −8.35691896663, −7.25047783803, −6.84552480603, −5.33985014788, −4.13559084051, −3.62482887082,
3.62482887082, 4.13559084051, 5.33985014788, 6.84552480603, 7.25047783803, 8.35691896663, 8.67236519668, 9.72466121073, 10.5195197796, 11.3731892768, 11.5559508175, 12.6521323881, 13.1937925053, 13.4821859369, 14.5135323404, 15.2472562838, 15.8196150869, 15.8734598001, 16.8741179353, 17.1598241820, 18.4192282680, 18.4485210975, 19.2406986299, 19.6678488176, 19.8970146357
