| L(s) = 1 | − 3·2-s + 6·4-s − 7-s − 10·8-s + 3·9-s + 11-s + 3·14-s + 15·16-s − 17-s − 9·18-s + 19-s − 3·22-s − 23-s + 3·25-s − 6·28-s + 29-s − 31-s − 21·32-s + 3·34-s + 18·36-s + 37-s − 3·38-s − 41-s + 43-s + 6·44-s + 3·46-s − 9·50-s + ⋯ |
| L(s) = 1 | − 3·2-s + 6·4-s − 7-s − 10·8-s + 3·9-s + 11-s + 3·14-s + 15·16-s − 17-s − 9·18-s + 19-s − 3·22-s − 23-s + 3·25-s − 6·28-s + 29-s − 31-s − 21·32-s + 3·34-s + 18·36-s + 37-s − 3·38-s − 41-s + 43-s + 6·44-s + 3·46-s − 9·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4973507331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4973507331\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 251 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 59 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 61 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.587089118816356606508794573300, −7.943906029771986606418082809963, −7.85890243121670900242281729721, −7.60770011685702659017541070939, −7.19308054045805578545423018584, −7.06193118339918372458285833476, −6.85032389312096640173458943905, −6.62863881728536657951153311396, −6.60059441440115932412898267055, −6.41737218706723170672936707924, −5.66078732644009157287456966480, −5.65734402908629410094255396023, −5.29511286539865447909580420384, −4.56826994551210916486967403356, −4.37964919113387758996924017437, −4.17007871439038847074469495222, −3.54462451090239104262059073847, −3.36453910786621113678447280529, −3.07884610974690439801936395900, −2.58292251964824413891401667894, −2.11344967430654172826563837789, −2.02916724052185377498976035686, −1.28192959638677343639575011895, −1.15651804087806272246957086694, −0.875377605188674701690736104607,
0.875377605188674701690736104607, 1.15651804087806272246957086694, 1.28192959638677343639575011895, 2.02916724052185377498976035686, 2.11344967430654172826563837789, 2.58292251964824413891401667894, 3.07884610974690439801936395900, 3.36453910786621113678447280529, 3.54462451090239104262059073847, 4.17007871439038847074469495222, 4.37964919113387758996924017437, 4.56826994551210916486967403356, 5.29511286539865447909580420384, 5.65734402908629410094255396023, 5.66078732644009157287456966480, 6.41737218706723170672936707924, 6.60059441440115932412898267055, 6.62863881728536657951153311396, 6.85032389312096640173458943905, 7.06193118339918372458285833476, 7.19308054045805578545423018584, 7.60770011685702659017541070939, 7.85890243121670900242281729721, 7.943906029771986606418082809963, 8.587089118816356606508794573300
