L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s − 2·11-s − 12-s + 2·14-s + 16-s − 5·17-s + 4·19-s + 2·21-s + 2·22-s − 2·23-s + 24-s + 2·25-s + 4·27-s − 2·28-s + 8·29-s + 2·31-s − 32-s + 2·33-s + 5·34-s − 8·37-s − 4·38-s + 4·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 0.603·11-s − 0.288·12-s + 0.534·14-s + 1/4·16-s − 1.21·17-s + 0.917·19-s + 0.436·21-s + 0.426·22-s − 0.417·23-s + 0.204·24-s + 2/5·25-s + 0.769·27-s − 0.377·28-s + 1.48·29-s + 0.359·31-s − 0.176·32-s + 0.348·33-s + 0.857·34-s − 1.31·37-s − 0.648·38-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3078707906\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3078707906\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 166 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−19.7456619493, −19.2340384236, −18.5811365388, −17.9933882063, −17.6203558362, −17.0792170868, −16.2322617871, −15.8316837652, −15.6362383357, −14.5486772983, −13.8355168830, −13.1753817804, −12.3989857266, −11.9099032971, −11.0931410729, −10.4538652996, −9.93956799794, −9.02514647032, −8.40788928395, −7.34350018754, −6.63216873792, −5.82717930465, −4.65562154609, −2.92100868367,
2.92100868367, 4.65562154609, 5.82717930465, 6.63216873792, 7.34350018754, 8.40788928395, 9.02514647032, 9.93956799794, 10.4538652996, 11.0931410729, 11.9099032971, 12.3989857266, 13.1753817804, 13.8355168830, 14.5486772983, 15.6362383357, 15.8316837652, 16.2322617871, 17.0792170868, 17.6203558362, 17.9933882063, 18.5811365388, 19.2340384236, 19.7456619493