| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s − 2·17-s + 4·19-s − 22-s − 24-s + 27-s − 33-s + 2·34-s − 4·38-s − 2·41-s + 43-s + 48-s − 49-s + 2·51-s − 54-s − 4·57-s + 59-s + 64-s + 66-s − 2·67-s + 4·73-s − 81-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s − 2·17-s + 4·19-s − 22-s − 24-s + 27-s − 33-s + 2·34-s − 4·38-s − 2·41-s + 43-s + 48-s − 49-s + 2·51-s − 54-s − 4·57-s + 59-s + 64-s + 66-s − 2·67-s + 4·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4557733678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4557733678\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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
−9.640438578146015356095088056803, −9.186739840430260288702629612261, −9.052828197951982238414009906431, −8.564377193879377025029768840084, −8.224625611435152696121861791144, −7.60397149181399659920888277447, −7.28232565099773069713868721022, −7.06814921452909538136307485302, −6.40933717768054318486031319926, −6.30639808719247771770436192254, −5.57494390043287875995783801860, −5.14074326690164370213115559887, −4.89899552314441338861927351671, −4.53848517681511873929800462976, −3.64547040969726651269707417843, −3.51672202746148954233044226584, −2.73465657168339997348935764546, −1.97074396961938038664392526891, −1.28771373136281373026529088091, −0.74476094257423663235645162715,
0.74476094257423663235645162715, 1.28771373136281373026529088091, 1.97074396961938038664392526891, 2.73465657168339997348935764546, 3.51672202746148954233044226584, 3.64547040969726651269707417843, 4.53848517681511873929800462976, 4.89899552314441338861927351671, 5.14074326690164370213115559887, 5.57494390043287875995783801860, 6.30639808719247771770436192254, 6.40933717768054318486031319926, 7.06814921452909538136307485302, 7.28232565099773069713868721022, 7.60397149181399659920888277447, 8.224625611435152696121861791144, 8.564377193879377025029768840084, 9.052828197951982238414009906431, 9.186739840430260288702629612261, 9.640438578146015356095088056803
