| L(s) = 1 | − 3-s − 4-s + 5-s − 11-s + 12-s − 15-s − 20-s − 2·23-s + 25-s + 27-s + 31-s + 33-s − 2·37-s + 44-s + 47-s − 49-s − 2·53-s − 55-s + 59-s + 60-s + 64-s + 67-s + 2·69-s − 2·71-s − 75-s − 81-s + 4·89-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 5-s − 11-s + 12-s − 15-s − 20-s − 2·23-s + 25-s + 27-s + 31-s + 33-s − 2·37-s + 44-s + 47-s − 49-s − 2·53-s − 55-s + 59-s + 60-s + 64-s + 67-s + 2·69-s − 2·71-s − 75-s − 81-s + 4·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2279293891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2279293891\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−14.08413891845906402596993050808, −14.04537565277773264555820701746, −13.21712899441185516945422273870, −13.09202432732099513747878414976, −12.17300394541636825118828706018, −12.03143578339260485688789095758, −11.18660620224420896302432564705, −10.56827862765709099156355649704, −10.16722206896139673314314510775, −9.771058824857733952186506670259, −8.995618197728112279657030086600, −8.495025006313518922315157726615, −7.900036263668485527728149050920, −6.97978145828428605996788480269, −6.13008093900409168117416047022, −5.87745731392387153437427664513, −4.86750891631774348547014659670, −4.83958004195883381465049249391, −3.48574567035271635281071490245, −2.19749917619495695658025058344,
2.19749917619495695658025058344, 3.48574567035271635281071490245, 4.83958004195883381465049249391, 4.86750891631774348547014659670, 5.87745731392387153437427664513, 6.13008093900409168117416047022, 6.97978145828428605996788480269, 7.900036263668485527728149050920, 8.495025006313518922315157726615, 8.995618197728112279657030086600, 9.771058824857733952186506670259, 10.16722206896139673314314510775, 10.56827862765709099156355649704, 11.18660620224420896302432564705, 12.03143578339260485688789095758, 12.17300394541636825118828706018, 13.09202432732099513747878414976, 13.21712899441185516945422273870, 14.04537565277773264555820701746, 14.08413891845906402596993050808
