L(s) = 1 | − 2-s − 3-s + 6-s − 7-s − 8-s − 3·9-s + 2·11-s − 4·13-s + 14-s − 16-s + 2·17-s + 3·18-s + 2·19-s + 21-s − 2·22-s + 24-s + 25-s + 4·26-s + 4·27-s − 4·31-s + 6·32-s − 2·33-s − 2·34-s − 2·38-s + 4·39-s − 2·41-s − 42-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 9-s + 0.603·11-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.458·19-s + 0.218·21-s − 0.426·22-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.718·31-s + 1.06·32-s − 0.348·33-s − 0.342·34-s − 0.324·38-s + 0.640·39-s − 0.312·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4348717527\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4348717527\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 2729 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 30 T + p T^{2} ) \) |
good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 124 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−14.4200891625, −14.1027899805, −13.5755130280, −13.0632400672, −12.3288857243, −12.1714781741, −11.7413776275, −11.2840121038, −10.8376388633, −10.2030550248, −9.81799193242, −9.29468420383, −8.92799057888, −8.57724878043, −7.78781484199, −7.36828044216, −6.80633575193, −6.04800116287, −5.89681708959, −5.03934054377, −4.61752056096, −3.55990490394, −2.98971237064, −2.11116110046, −0.589216363430,
0.589216363430, 2.11116110046, 2.98971237064, 3.55990490394, 4.61752056096, 5.03934054377, 5.89681708959, 6.04800116287, 6.80633575193, 7.36828044216, 7.78781484199, 8.57724878043, 8.92799057888, 9.29468420383, 9.81799193242, 10.2030550248, 10.8376388633, 11.2840121038, 11.7413776275, 12.1714781741, 12.3288857243, 13.0632400672, 13.5755130280, 14.1027899805, 14.4200891625