| L(s) = 1 | − 3-s + 2·5-s + 7-s + 3·9-s − 3·11-s + 2·13-s − 2·15-s + 3·17-s + 7·19-s − 21-s + 3·23-s + 3·25-s − 8·27-s − 3·29-s − 8·31-s + 3·33-s + 2·35-s + 7·37-s − 2·39-s + 9·41-s − 11·43-s + 6·45-s + 7·49-s − 3·51-s − 12·53-s − 6·55-s − 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 9-s − 0.904·11-s + 0.554·13-s − 0.516·15-s + 0.727·17-s + 1.60·19-s − 0.218·21-s + 0.625·23-s + 3/5·25-s − 1.53·27-s − 0.557·29-s − 1.43·31-s + 0.522·33-s + 0.338·35-s + 1.15·37-s − 0.320·39-s + 1.40·41-s − 1.67·43-s + 0.894·45-s + 49-s − 0.420·51-s − 1.64·53-s − 0.809·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577838413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577838413\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−12.31543963923187685107399222014, −11.57063720767422974062168494641, −11.25450457241074693523981802724, −10.87247590479960051581959150199, −10.35156453584152564180173308457, −9.775142104213168411996169639242, −9.505655445924025053132115558918, −9.121818417072777770932432991858, −8.210209327450871420345092967992, −7.73438706040159902965256644242, −7.33093914032845698701516969171, −6.83210411564571409047157481953, −5.88413821826063427914267247536, −5.73676857566534083124455674535, −5.15902425992894204162895801001, −4.62920393158057106026919346181, −3.71086553297806021306157772874, −3.04107735360159192429699392409, −1.97844682059782919753617350785, −1.17743233124008280959897587172,
1.17743233124008280959897587172, 1.97844682059782919753617350785, 3.04107735360159192429699392409, 3.71086553297806021306157772874, 4.62920393158057106026919346181, 5.15902425992894204162895801001, 5.73676857566534083124455674535, 5.88413821826063427914267247536, 6.83210411564571409047157481953, 7.33093914032845698701516969171, 7.73438706040159902965256644242, 8.210209327450871420345092967992, 9.121818417072777770932432991858, 9.505655445924025053132115558918, 9.775142104213168411996169639242, 10.35156453584152564180173308457, 10.87247590479960051581959150199, 11.25450457241074693523981802724, 11.57063720767422974062168494641, 12.31543963923187685107399222014
