| L(s) = 1 | − 2-s + 4-s + 7-s − 3·8-s + 9-s − 11-s − 14-s + 16-s + 2·17-s − 18-s − 5·19-s + 22-s + 23-s − 6·25-s + 28-s − 3·29-s + 4·31-s + 32-s − 2·34-s + 36-s − 2·37-s + 5·38-s − 2·41-s + 3·43-s − 44-s − 46-s − 5·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.14·19-s + 0.213·22-s + 0.208·23-s − 6/5·25-s + 0.188·28-s − 0.557·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + 0.811·38-s − 0.312·41-s + 0.457·43-s − 0.150·44-s − 0.147·46-s − 5/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2043 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2043 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4850836291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4850836291\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 227 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 3 T - 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T - 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−18.8218039111, −18.2556014121, −17.7984239046, −17.2734130001, −16.8920017817, −16.0439766146, −15.6664614494, −15.1260876193, −14.6250145220, −13.9143378183, −13.2206502856, −12.5790872549, −12.0074027484, −11.3363162516, −10.8738967347, −9.96847666748, −9.64317423446, −8.72329847974, −8.23914418501, −7.50221257179, −6.63881331710, −5.95403234969, −4.92142252017, −3.71405741145, −2.26563134296,
2.26563134296, 3.71405741145, 4.92142252017, 5.95403234969, 6.63881331710, 7.50221257179, 8.23914418501, 8.72329847974, 9.64317423446, 9.96847666748, 10.8738967347, 11.3363162516, 12.0074027484, 12.5790872549, 13.2206502856, 13.9143378183, 14.6250145220, 15.1260876193, 15.6664614494, 16.0439766146, 16.8920017817, 17.2734130001, 17.7984239046, 18.2556014121, 18.8218039111
