| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 2·7-s + 4·8-s + 2·9-s − 4·10-s + 6·12-s + 4·13-s + 4·14-s − 4·15-s + 5·16-s + 4·18-s − 10·19-s − 6·20-s + 4·21-s + 8·24-s − 25-s + 8·26-s + 6·27-s + 6·28-s − 6·29-s − 8·30-s + 14·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 2/3·9-s − 1.26·10-s + 1.73·12-s + 1.10·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.942·18-s − 2.29·19-s − 1.34·20-s + 0.872·21-s + 1.63·24-s − 1/5·25-s + 1.56·26-s + 1.15·27-s + 1.13·28-s − 1.11·29-s − 1.46·30-s + 2.51·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.383813821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.383813821\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.51681049479976715515495458736, −11.43854637309066998618799233508, −10.72214498184088405583760294808, −10.60314747515121830693942006683, −9.913561959008795010014553711726, −9.173798326193050005880679000962, −8.550678622946176948223109707713, −8.309662155208741088838408339798, −7.968092296445966825260904727649, −7.49965213761600358516025530418, −6.61342180482473715016656383816, −6.49397020060739909292475709936, −5.84108691251726407248319257508, −4.99367285908152138982104726730, −4.46225439661671786023374444253, −4.10728915493454226033796375498, −3.64159708286768808345882270281, −2.91408694033091515323513245375, −2.32661717346622627370675610005, −1.48399474842316005654978594715,
1.48399474842316005654978594715, 2.32661717346622627370675610005, 2.91408694033091515323513245375, 3.64159708286768808345882270281, 4.10728915493454226033796375498, 4.46225439661671786023374444253, 4.99367285908152138982104726730, 5.84108691251726407248319257508, 6.49397020060739909292475709936, 6.61342180482473715016656383816, 7.49965213761600358516025530418, 7.968092296445966825260904727649, 8.309662155208741088838408339798, 8.550678622946176948223109707713, 9.173798326193050005880679000962, 9.913561959008795010014553711726, 10.60314747515121830693942006683, 10.72214498184088405583760294808, 11.43854637309066998618799233508, 11.51681049479976715515495458736
