| L(s) = 1 | + 56·9-s − 12·11-s + 304·19-s + 936·29-s − 536·31-s − 148·41-s − 147·49-s + 2.17e3·59-s + 348·61-s − 2.11e3·71-s − 724·79-s + 2.52e3·81-s − 316·89-s − 672·99-s − 1.93e3·101-s − 2.24e3·109-s − 2.16e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.04e3·169-s + ⋯ |
| L(s) = 1 | + 2.07·9-s − 0.328·11-s + 3.67·19-s + 5.99·29-s − 3.10·31-s − 0.563·41-s − 3/7·49-s + 4.80·59-s + 0.730·61-s − 3.53·71-s − 1.03·79-s + 3.45·81-s − 0.376·89-s − 0.682·99-s − 1.90·101-s − 1.96·109-s − 1.62·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.75·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(32.59570710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(32.59570710\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| good | 3 | \( 1 - 56 T^{2} + 616 T^{4} + 9082 T^{6} + 616 p^{6} T^{8} - 56 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( ( 1 + 6 T + 1138 T^{2} - 8648 T^{3} + 1138 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 13 | \( 1 - 6048 T^{2} + 21878208 T^{4} - 346391686 p^{2} T^{6} + 21878208 p^{6} T^{8} - 6048 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 16200 T^{2} + 128022712 T^{4} - 712966987550 T^{6} + 128022712 p^{6} T^{8} - 16200 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 - 8 p T + 6157 T^{2} + 2640 T^{3} + 6157 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 28994 T^{2} + 654665999 T^{4} - 8788260020284 T^{6} + 654665999 p^{6} T^{8} - 28994 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 468 T + 112852 T^{2} - 19590970 T^{3} + 112852 p^{3} T^{4} - 468 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 + 268 T + 3251 p T^{2} + 15045992 T^{3} + 3251 p^{4} T^{4} + 268 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 129634 T^{2} + 11754341511 T^{4} - 663381932115452 T^{6} + 11754341511 p^{6} T^{8} - 129634 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 74 T + 148867 T^{2} + 6978388 T^{3} + 148867 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 187450 T^{2} + 6750681495 T^{4} + 303916224808084 T^{6} + 6750681495 p^{6} T^{8} - 187450 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 341936 T^{2} + 57108904064 T^{4} - 6652664760342206 T^{6} + 57108904064 p^{6} T^{8} - 341936 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 723658 T^{2} + 237614752663 T^{4} - 45237793900217804 T^{6} + 237614752663 p^{6} T^{8} - 723658 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 - 1088 T + 930505 T^{2} - 466405760 T^{3} + 930505 p^{3} T^{4} - 1088 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 - 174 T + 132487 T^{2} + 108451156 T^{3} + 132487 p^{3} T^{4} - 174 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 1153906 T^{2} + 648182674167 T^{4} - 233822456055518108 T^{6} + 648182674167 p^{6} T^{8} - 1153906 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 1056 T + 831813 T^{2} + 484957632 T^{3} + 831813 p^{3} T^{4} + 1056 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 1403114 T^{2} + 989693154207 T^{4} - 456630565235511372 T^{6} + 989693154207 p^{6} T^{8} - 1403114 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 + 362 T + 559502 T^{2} - 5683900 T^{3} + 559502 p^{3} T^{4} + 362 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 2018210 T^{2} + 1903802939287 T^{4} - 1223243302094648380 T^{6} + 1903802939287 p^{6} T^{8} - 2018210 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 158 T + 258707 T^{2} - 637693636 T^{3} + 258707 p^{3} T^{4} + 158 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 238728 T^{2} + 1731919551432 T^{4} + 586934206071184834 T^{6} + 1731919551432 p^{6} T^{8} + 238728 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−4.57867913182151299037358885079, −4.43232960918880243217376785179, −4.30564635742914747353174194393, −4.14944000407540322859523325705, −4.05855253932307672151631162564, −3.85484056156119603248733157109, −3.68463240442141048186566397110, −3.50413919354040086884177753166, −3.23404153586739173811453191441, −3.05495178338149961446078713176, −2.93959942554707952747368328309, −2.92045792091480441787885959158, −2.71717980158731758251328880418, −2.56534617947940068938261321180, −2.24060772810250360738799240702, −1.85571276752855392009783463179, −1.82455701481935398673288674595, −1.61547476316664241725605203464, −1.51231779052126995798521566895, −1.23226243063778279356576602426, −0.958272224876014019031163659981, −0.798407358131982002676912277793, −0.69545256805074627351411815136, −0.58379087599645976833308322536, −0.29210955717614503922550583058,
0.29210955717614503922550583058, 0.58379087599645976833308322536, 0.69545256805074627351411815136, 0.798407358131982002676912277793, 0.958272224876014019031163659981, 1.23226243063778279356576602426, 1.51231779052126995798521566895, 1.61547476316664241725605203464, 1.82455701481935398673288674595, 1.85571276752855392009783463179, 2.24060772810250360738799240702, 2.56534617947940068938261321180, 2.71717980158731758251328880418, 2.92045792091480441787885959158, 2.93959942554707952747368328309, 3.05495178338149961446078713176, 3.23404153586739173811453191441, 3.50413919354040086884177753166, 3.68463240442141048186566397110, 3.85484056156119603248733157109, 4.05855253932307672151631162564, 4.14944000407540322859523325705, 4.30564635742914747353174194393, 4.43232960918880243217376785179, 4.57867913182151299037358885079
Plot not available for L-functions of degree greater than 10.
