| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 13-s + 2·19-s − 25-s − 4·27-s + 2·31-s − 3·36-s + 37-s − 2·39-s + 43-s − 52-s − 4·57-s + 2·61-s + 64-s + 4·67-s − 73-s + 2·75-s − 2·76-s − 2·79-s + 5·81-s − 4·93-s − 97-s + 100-s − 103-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 13-s + 2·19-s − 25-s − 4·27-s + 2·31-s − 3·36-s + 37-s − 2·39-s + 43-s − 52-s − 4·57-s + 2·61-s + 64-s + 4·67-s − 73-s + 2·75-s − 2·76-s − 2·79-s + 5·81-s − 4·93-s − 97-s + 100-s − 103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5694052738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5694052738\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.662533519613406673025375450039, −9.656256200553710211661201314125, −8.712823375299719566647542485077, −8.519721967645849578794199676846, −7.967863811140132526758333396627, −7.59237135916246935726521461125, −7.10471513391789749006308749361, −6.78439202537346547533396074802, −6.30802812004285915563331866730, −5.91798277993836865558801516530, −5.47902718184964787292789826781, −5.36045434435459453431243724307, −4.68075724484927963861264744053, −4.49543654635284036209384951782, −3.82984864835340156491718341275, −3.75096762742559148294459964529, −2.81096549727465753323732009727, −2.02927544728911295966570363215, −1.02725645095688147557234173972, −0.885188284368538904412591531769,
0.885188284368538904412591531769, 1.02725645095688147557234173972, 2.02927544728911295966570363215, 2.81096549727465753323732009727, 3.75096762742559148294459964529, 3.82984864835340156491718341275, 4.49543654635284036209384951782, 4.68075724484927963861264744053, 5.36045434435459453431243724307, 5.47902718184964787292789826781, 5.91798277993836865558801516530, 6.30802812004285915563331866730, 6.78439202537346547533396074802, 7.10471513391789749006308749361, 7.59237135916246935726521461125, 7.967863811140132526758333396627, 8.519721967645849578794199676846, 8.712823375299719566647542485077, 9.656256200553710211661201314125, 9.662533519613406673025375450039
