| L(s) = 1 | − 2·2-s − 5·3-s − 3·5-s + 10·6-s − 64·7-s + 8·8-s + 27·9-s + 6·10-s + 8·11-s + 69·13-s + 128·14-s + 15·15-s − 16·16-s − 19·17-s − 54·18-s + 152·19-s + 320·21-s − 16·22-s − 67·23-s − 40·24-s + 125·25-s − 138·26-s − 280·27-s − 51·29-s − 30·30-s − 264·31-s − 40·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s − 0.268·5-s + 0.680·6-s − 3.45·7-s + 0.353·8-s + 9-s + 0.189·10-s + 0.219·11-s + 1.47·13-s + 2.44·14-s + 0.258·15-s − 1/4·16-s − 0.271·17-s − 0.707·18-s + 1.83·19-s + 3.32·21-s − 0.155·22-s − 0.607·23-s − 0.340·24-s + 25-s − 1.04·26-s − 1.99·27-s − 0.326·29-s − 0.182·30-s − 1.52·31-s − 0.211·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3549217298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3549217298\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 69 T + 2564 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 19 T - 4552 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 67 T - 7678 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 51 T - 21788 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 132 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 413 T + 101648 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 3 p T - 34 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 617 T + 276866 T^{2} - 617 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 383 T - 2188 T^{2} + 383 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 599 T + 153422 T^{2} - 599 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 217 T - 179892 T^{2} - 217 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 225 T - 250138 T^{2} - 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 701 T + 133490 T^{2} + 701 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 1015 T + 641208 T^{2} + 1015 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 349 T - 371238 T^{2} + 349 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 592 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1349 T + 1114832 T^{2} - 1349 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 613 T - 536904 T^{2} - 613 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−16.10441543491617007622034977627, −15.84771353709858655258013640870, −15.60107345727530123403295476909, −14.23036720749007182959031374912, −13.39787648912094679968147737759, −12.91692362915614939792206810596, −12.79845573107374331924032903380, −11.83387670664792377248638250551, −11.20816321064741982072667981049, −10.30868495344333800147467302640, −9.981711083431891175352212229036, −9.118812386362574629286449912190, −9.107689955945650265477429993987, −7.35645399658817551427577787639, −7.06386732334685446277813405569, −6.00150474306587485868330129324, −5.85081849213172110714434792469, −3.91136547787382334409460444903, −3.32044710855048030118575657231, −0.60970772749884074980221549934,
0.60970772749884074980221549934, 3.32044710855048030118575657231, 3.91136547787382334409460444903, 5.85081849213172110714434792469, 6.00150474306587485868330129324, 7.06386732334685446277813405569, 7.35645399658817551427577787639, 9.107689955945650265477429993987, 9.118812386362574629286449912190, 9.981711083431891175352212229036, 10.30868495344333800147467302640, 11.20816321064741982072667981049, 11.83387670664792377248638250551, 12.79845573107374331924032903380, 12.91692362915614939792206810596, 13.39787648912094679968147737759, 14.23036720749007182959031374912, 15.60107345727530123403295476909, 15.84771353709858655258013640870, 16.10441543491617007622034977627
