L(s) = 1 | − 2·3-s − 4-s + 7-s − 2·8-s − 2·9-s + 4·11-s + 2·12-s − 4·13-s + 16-s + 6·17-s + 2·19-s − 2·21-s + 8·23-s + 4·24-s − 10·25-s + 10·27-s − 28-s − 4·29-s − 4·31-s + 4·32-s − 8·33-s + 2·36-s − 4·37-s + 8·39-s + 6·41-s − 4·43-s − 4·44-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.377·7-s − 0.707·8-s − 2/3·9-s + 1.20·11-s + 0.577·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s + 0.816·24-s − 2·25-s + 1.92·27-s − 0.188·28-s − 0.742·29-s − 0.718·31-s + 0.707·32-s − 1.39·33-s + 1/3·36-s − 0.657·37-s + 1.28·39-s + 0.937·41-s − 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3192126330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3192126330\) |
\(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.8926107789, −19.5800270198, −18.9333083528, −18.2120541317, −17.5895382325, −17.2128532162, −16.9141484172, −16.3370810014, −15.2564971903, −14.6077730657, −14.3678900325, −13.5582912652, −12.3052609901, −12.2794334476, −11.3085026503, −11.2313614144, −9.76554711946, −9.48952508504, −8.49812018178, −7.57571100089, −6.47803659589, −5.57928681743, −5.08673463815, −3.45773984942,
3.45773984942, 5.08673463815, 5.57928681743, 6.47803659589, 7.57571100089, 8.49812018178, 9.48952508504, 9.76554711946, 11.2313614144, 11.3085026503, 12.2794334476, 12.3052609901, 13.5582912652, 14.3678900325, 14.6077730657, 15.2564971903, 16.3370810014, 16.9141484172, 17.2128532162, 17.5895382325, 18.2120541317, 18.9333083528, 19.5800270198, 19.8926107789