| L(s) = 1 | − 2·3-s − 2·5-s − 2·9-s + 4·11-s + 2·13-s + 4·15-s − 8·17-s − 6·19-s − 8·23-s + 2·25-s + 10·27-s + 8·29-s − 12·31-s − 8·33-s − 8·37-s − 4·39-s − 4·43-s + 4·45-s + 12·47-s + 49-s + 16·51-s − 4·53-s − 8·55-s + 12·57-s − 6·59-s − 2·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 2/3·9-s + 1.20·11-s + 0.554·13-s + 1.03·15-s − 1.94·17-s − 1.37·19-s − 1.66·23-s + 2/5·25-s + 1.92·27-s + 1.48·29-s − 2.15·31-s − 1.39·33-s − 1.31·37-s − 0.640·39-s − 0.609·43-s + 0.596·45-s + 1.75·47-s + 1/7·49-s + 2.24·51-s − 0.549·53-s − 1.07·55-s + 1.58·57-s − 0.781·59-s − 0.256·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−16.8376077472, −16.0039625329, −15.5421032288, −15.4659592002, −14.4247744515, −14.2069489001, −13.7979042634, −12.6775996869, −12.6607831742, −11.8923297977, −11.4050949010, −11.3235241291, −10.6358774624, −10.2732206713, −9.11789403464, −8.60522905669, −8.56060780623, −7.48523560916, −6.67949194679, −6.34948564084, −5.82613210142, −4.88658861408, −4.18465932716, −3.61118272832, −2.14556791802, 0,
2.14556791802, 3.61118272832, 4.18465932716, 4.88658861408, 5.82613210142, 6.34948564084, 6.67949194679, 7.48523560916, 8.56060780623, 8.60522905669, 9.11789403464, 10.2732206713, 10.6358774624, 11.3235241291, 11.4050949010, 11.8923297977, 12.6607831742, 12.6775996869, 13.7979042634, 14.2069489001, 14.4247744515, 15.4659592002, 15.5421032288, 16.0039625329, 16.8376077472
