| L(s) = 1 | − 3·9-s + 6·29-s + 6·41-s + 64-s + 6·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 3·9-s + 6·29-s + 6·41-s + 64-s + 6·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.309423322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309423322\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 29 | \( ( 1 - T )^{6} \) |
| good | 2 | \( 1 - T^{6} + T^{12} \) |
| 7 | \( 1 - T^{6} + T^{12} \) |
| 11 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 13 | \( 1 - T^{6} + T^{12} \) |
| 17 | \( 1 - T^{6} + T^{12} \) |
| 19 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 23 | \( ( 1 + T^{2} )^{6} \) |
| 31 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 37 | \( ( 1 + T^{2} )^{6} \) |
| 41 | \( ( 1 - T + T^{2} )^{6} \) |
| 43 | \( ( 1 + T^{2} )^{6} \) |
| 47 | \( 1 - T^{6} + T^{12} \) |
| 53 | \( ( 1 + T^{2} )^{6} \) |
| 59 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 61 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 67 | \( 1 - T^{6} + T^{12} \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( ( 1 + T^{2} )^{6} \) |
| 79 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 83 | \( ( 1 + T^{2} )^{6} \) |
| 89 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 97 | \( ( 1 + T^{2} )^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.91249079498786374679198883511, −4.77184252945360427516941172874, −4.70964438917740111320191090345, −4.69381539196767019473620969287, −4.23406977267204944449113408011, −4.22692813553507140304874873045, −4.01880319095264500850401484173, −3.85693782182176314157368440837, −3.82006731852525174849727657097, −3.44912402999377911778468326972, −3.25624927978870964100807954002, −3.17888835556466171006101325012, −3.16310656184559475310269913081, −2.66794085219172610065449103803, −2.63978715530605962633459164257, −2.51474962277231355101912560480, −2.46164880092221341098034529901, −2.39482021786442359122130640401, −2.38481037981051107665735447477, −1.87237842216213082542263440373, −1.35548121006515235899879292803, −1.16097646635749246192058199182, −1.14951364616999161823297943983, −0.811823675829709997785431308179, −0.59307221157721873904594030647,
0.59307221157721873904594030647, 0.811823675829709997785431308179, 1.14951364616999161823297943983, 1.16097646635749246192058199182, 1.35548121006515235899879292803, 1.87237842216213082542263440373, 2.38481037981051107665735447477, 2.39482021786442359122130640401, 2.46164880092221341098034529901, 2.51474962277231355101912560480, 2.63978715530605962633459164257, 2.66794085219172610065449103803, 3.16310656184559475310269913081, 3.17888835556466171006101325012, 3.25624927978870964100807954002, 3.44912402999377911778468326972, 3.82006731852525174849727657097, 3.85693782182176314157368440837, 4.01880319095264500850401484173, 4.22692813553507140304874873045, 4.23406977267204944449113408011, 4.69381539196767019473620969287, 4.70964438917740111320191090345, 4.77184252945360427516941172874, 4.91249079498786374679198883511
Plot not available for L-functions of degree greater than 10.
