| L(s) = 1 | + 21·19-s − 33·37-s + 8·64-s + 21·73-s + 12·109-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 | + 4.81·19-s − 5.42·37-s + 64-s + 2.45·73-s + 1.14·109-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 + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377878341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377878341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 5 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 17 T^{3} + p^{3} T^{6} )( 1 + 37 T^{3} + p^{3} T^{6} ) \) |
| 11 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{3}( 1 + T + p T^{2} )^{3} \) |
| 23 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 19 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} ) \) |
| 37 | \( ( 1 + T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \) |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 449 T^{3} + p^{3} T^{6} )( 1 - 71 T^{3} + p^{3} T^{6} ) \) |
| 47 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( ( 1 + p T^{2} )^{6} \) |
| 59 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 719 T^{3} + p^{3} T^{6} )( 1 + 901 T^{3} + p^{3} T^{6} ) \) |
| 67 | \( ( 1 - 1007 T^{3} + p^{3} T^{6} )( 1 + 127 T^{3} + p^{3} T^{6} ) \) |
| 71 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 73 | \( ( 1 - 17 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 - 503 T^{3} + p^{3} T^{6} )( 1 + 1387 T^{3} + p^{3} T^{6} ) \) |
| 83 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 97 | \( ( 1 - 1853 T^{3} + p^{3} T^{6} )( 1 + 523 T^{3} + p^{3} T^{6} ) \) |
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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
−5.39693270257026677572625769455, −5.19740217125182222965565813465, −5.15651556945074164642761987560, −5.11962502753319739703464465246, −5.02359505910417287553351515004, −4.90893828520675537275892166809, −4.75073829087518742391180276142, −4.35666792485979001349701728828, −3.89218635874035959723193267001, −3.89140524968830114346084977688, −3.62090610091713417808477378267, −3.61496372560854211377731395538, −3.57735746289543123429592203540, −3.36275353103461006961630319987, −3.06786296204435527771850437524, −2.72184952383379176943349457570, −2.60294011361708084539319257101, −2.55910998280373179008796778097, −2.23758673685542541862360751672, −1.65923592381617382130211192189, −1.53827353629654300169931148094, −1.50235869957726741863936480270, −1.21879467525710883604211949684, −0.71330027533948425462725648783, −0.37773775011047904446030288029,
0.37773775011047904446030288029, 0.71330027533948425462725648783, 1.21879467525710883604211949684, 1.50235869957726741863936480270, 1.53827353629654300169931148094, 1.65923592381617382130211192189, 2.23758673685542541862360751672, 2.55910998280373179008796778097, 2.60294011361708084539319257101, 2.72184952383379176943349457570, 3.06786296204435527771850437524, 3.36275353103461006961630319987, 3.57735746289543123429592203540, 3.61496372560854211377731395538, 3.62090610091713417808477378267, 3.89140524968830114346084977688, 3.89218635874035959723193267001, 4.35666792485979001349701728828, 4.75073829087518742391180276142, 4.90893828520675537275892166809, 5.02359505910417287553351515004, 5.11962502753319739703464465246, 5.15651556945074164642761987560, 5.19740217125182222965565813465, 5.39693270257026677572625769455
Plot not available for L-functions of degree greater than 10.
