| L(s) = 1 | + 2-s + 3·5-s − 7-s + 8-s − 2·9-s + 3·10-s + 3·11-s − 2·13-s − 14-s − 16-s − 2·18-s − 3·19-s + 3·22-s + 2·23-s + 5·25-s − 2·26-s − 3·27-s + 2·31-s − 6·32-s − 3·35-s + 37-s − 3·38-s + 3·40-s + 41-s − 9·43-s − 6·45-s + 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·5-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.904·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.471·18-s − 0.688·19-s + 0.639·22-s + 0.417·23-s + 25-s − 0.392·26-s − 0.577·27-s + 0.359·31-s − 1.06·32-s − 0.507·35-s + 0.164·37-s − 0.486·38-s + 0.474·40-s + 0.156·41-s − 1.37·43-s − 0.894·45-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16023 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16023 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.720293944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.720293944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 109 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 11 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 95 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 72 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 114 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 80 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−16.0352033901, −15.3432718935, −14.8799545848, −14.4384834765, −14.0086546361, −13.6187692223, −13.2770417603, −12.7456884173, −12.2361295868, −11.6780588755, −11.0490646565, −10.5819906272, −9.92951538494, −9.46555978019, −8.99096587120, −8.50157019385, −7.54132210151, −6.91342354778, −6.26834098700, −5.88390493282, −5.10456717154, −4.55550116523, −3.69246268756, −2.71158713242, −1.80992432674,
1.80992432674, 2.71158713242, 3.69246268756, 4.55550116523, 5.10456717154, 5.88390493282, 6.26834098700, 6.91342354778, 7.54132210151, 8.50157019385, 8.99096587120, 9.46555978019, 9.92951538494, 10.5819906272, 11.0490646565, 11.6780588755, 12.2361295868, 12.7456884173, 13.2770417603, 13.6187692223, 14.0086546361, 14.4384834765, 14.8799545848, 15.3432718935, 16.0352033901
