| L(s) = 1 | + 12·7-s + 8-s − 9·11-s + 8·27-s − 6·31-s − 60·37-s + 69·49-s + 12·56-s − 108·77-s − 9·83-s − 9·88-s − 6·103-s + 90·113-s + 60·121-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 + 96·189-s + ⋯ |
| L(s) = 1 | + 4.53·7-s + 0.353·8-s − 2.71·11-s + 1.53·27-s − 1.07·31-s − 9.86·37-s + 69/7·49-s + 1.60·56-s − 12.3·77-s − 0.987·83-s − 0.959·88-s − 0.591·103-s + 8.46·113-s + 5.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 6.98·189-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.248430539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248430539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 8 T^{3} + 37 T^{6} - 8 p^{3} T^{9} + p^{6} T^{12} \) |
| 5 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + T + p T^{2} )^{3} \) |
| 11 | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{3} \) |
| 13 | \( ( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 + 90 T^{3} + 3187 T^{6} + 90 p^{3} T^{9} + p^{6} T^{12} \) |
| 23 | \( 1 + 198 T^{3} + 27037 T^{6} + 198 p^{3} T^{9} + p^{6} T^{12} \) |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 37 | \( ( 1 + 10 T + p T^{2} )^{6} \) |
| 41 | \( 1 - 378 T^{3} + 73963 T^{6} - 378 p^{3} T^{9} + p^{6} T^{12} \) |
| 43 | \( 1 + 452 T^{3} + 124797 T^{6} + 452 p^{3} T^{9} + p^{6} T^{12} \) |
| 47 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( 1 - 738 T^{3} + 395767 T^{6} - 738 p^{3} T^{9} + p^{6} T^{12} \) |
| 59 | \( 1 + 864 T^{3} + 541117 T^{6} + 864 p^{3} T^{9} + p^{6} T^{12} \) |
| 61 | \( 1 + 668 T^{3} + 219243 T^{6} + 668 p^{3} T^{9} + p^{6} T^{12} \) |
| 67 | \( 1 + 1064 T^{3} + 831333 T^{6} + 1064 p^{3} T^{9} + p^{6} T^{12} \) |
| 71 | \( 1 + 1062 T^{3} + 769933 T^{6} + 1062 p^{3} T^{9} + p^{6} T^{12} \) |
| 73 | \( 1 + 218 T^{3} - 341493 T^{6} + 218 p^{3} T^{9} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 503 T^{3} + p^{3} T^{6} )( 1 + 1387 T^{3} + p^{3} T^{6} ) \) |
| 83 | \( ( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 - 1386 T^{3} + 1216027 T^{6} - 1386 p^{3} T^{9} + p^{6} T^{12} \) |
| 97 | \( 1 - 34 T^{3} - 911517 T^{6} - 34 p^{3} T^{9} + p^{6} T^{12} \) |
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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
−5.46988346462318930481884340639, −5.30188204568022075377504544624, −5.07509753623535944850096342668, −5.06657700635795102000246560351, −4.90281160968276419407991246234, −4.79656210946060467664478961336, −4.75262839185516036467787169067, −4.47276060416477687712057797388, −4.44507946815948658660868714569, −3.95005449273002420896842129601, −3.90504931861227729387431166062, −3.48521435193339866317299412278, −3.31272950090512979660215625329, −3.28096676694320416899684491907, −3.25649857975766699044802385393, −2.78604311538718250587655825895, −2.38506236960954075175712581519, −2.18196849201305948438451788655, −2.06880651795526619458696259499, −1.82552688225243442739535305241, −1.72588733642421442884050129811, −1.61717570057715997756820484829, −1.31984503617888689821392152302, −0.836217416795624948279877900830, −0.17712679405171862800941888204,
0.17712679405171862800941888204, 0.836217416795624948279877900830, 1.31984503617888689821392152302, 1.61717570057715997756820484829, 1.72588733642421442884050129811, 1.82552688225243442739535305241, 2.06880651795526619458696259499, 2.18196849201305948438451788655, 2.38506236960954075175712581519, 2.78604311538718250587655825895, 3.25649857975766699044802385393, 3.28096676694320416899684491907, 3.31272950090512979660215625329, 3.48521435193339866317299412278, 3.90504931861227729387431166062, 3.95005449273002420896842129601, 4.44507946815948658660868714569, 4.47276060416477687712057797388, 4.75262839185516036467787169067, 4.79656210946060467664478961336, 4.90281160968276419407991246234, 5.06657700635795102000246560351, 5.07509753623535944850096342668, 5.30188204568022075377504544624, 5.46988346462318930481884340639
Plot not available for L-functions of degree greater than 10.
