| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s + 2·7-s − 8-s − 2·9-s − 2·10-s + 4·11-s − 2·12-s − 2·13-s − 2·14-s − 4·15-s + 16-s + 2·18-s − 6·19-s + 2·20-s − 4·21-s − 4·22-s + 2·24-s − 6·25-s + 2·26-s + 10·27-s + 2·28-s + 4·30-s − 12·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.632·10-s + 1.20·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.471·18-s − 1.37·19-s + 0.447·20-s − 0.872·21-s − 0.852·22-s + 0.408·24-s − 6/5·25-s + 0.392·26-s + 1.92·27-s + 0.377·28-s + 0.730·30-s − 2.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3791316750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3791316750\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.5014287171, −18.2120541317, −18.1790379174, −17.3319169048, −17.2128532162, −16.9216980065, −16.3370810014, −15.4659592002, −14.6077730657, −14.4247744515, −13.7979042634, −12.6775996869, −12.3052609901, −11.4050949010, −11.2313614144, −10.6358774624, −9.76554711946, −9.11789403464, −8.56060780623, −7.57571100089, −6.67949194679, −5.82613210142, −5.57928681743, −4.18465932716, −2.14556791802,
2.14556791802, 4.18465932716, 5.57928681743, 5.82613210142, 6.67949194679, 7.57571100089, 8.56060780623, 9.11789403464, 9.76554711946, 10.6358774624, 11.2313614144, 11.4050949010, 12.3052609901, 12.6775996869, 13.7979042634, 14.4247744515, 14.6077730657, 15.4659592002, 16.3370810014, 16.9216980065, 17.2128532162, 17.3319169048, 18.1790379174, 18.2120541317, 19.5014287171
