| L(s) = 1 | − 2-s − 4-s − 3·5-s − 5·7-s + 3·8-s + 9-s + 3·10-s + 8·11-s + 4·13-s + 5·14-s − 16-s − 4·17-s − 18-s + 8·19-s + 3·20-s − 8·22-s + 2·25-s − 4·26-s + 5·28-s + 8·29-s − 4·31-s − 5·32-s + 4·34-s + 15·35-s − 36-s − 4·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.34·5-s − 1.88·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s + 2.41·11-s + 1.10·13-s + 1.33·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.83·19-s + 0.670·20-s − 1.70·22-s + 2/5·25-s − 0.784·26-s + 0.944·28-s + 1.48·29-s − 0.718·31-s − 0.883·32-s + 0.685·34-s + 2.53·35-s − 1/6·36-s − 0.657·37-s − 1.29·38-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
| \(L(1)\) |
\(\approx\) |
\(0.4723075322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4723075322\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 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
−16.4553288530, −16.0167583574, −15.8734598001, −15.4455527900, −14.5135323404, −14.1156694920, −13.5066773477, −13.1937925053, −12.4478600114, −11.9717989438, −11.5559508175, −11.0212342426, −10.1385531201, −9.72466121073, −9.26580898132, −8.67236519668, −8.42193331174, −7.25047783803, −6.97981434956, −6.46117181372, −5.52040688745, −4.13559084051, −3.95366310026, −3.30373640608, −1.02432023627,
1.02432023627, 3.30373640608, 3.95366310026, 4.13559084051, 5.52040688745, 6.46117181372, 6.97981434956, 7.25047783803, 8.42193331174, 8.67236519668, 9.26580898132, 9.72466121073, 10.1385531201, 11.0212342426, 11.5559508175, 11.9717989438, 12.4478600114, 13.1937925053, 13.5066773477, 14.1156694920, 14.5135323404, 15.4455527900, 15.8734598001, 16.0167583574, 16.4553288530
