| L(s) = 1 | − 24·19-s + 3·37-s + 8·64-s + 21·73-s + 102·109-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 | − 5.50·19-s + 0.493·37-s + 64-s + 2.45·73-s + 9.76·109-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 | \( ( 1 + T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \) |
| 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 | \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \) |
| 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
−5.59608336081598638679179934646, −5.31865282800980099572711364246, −5.28125687756670293508646874366, −5.07795171912515974949868654564, −4.69839459516157866001107771637, −4.61482243608506239092839840531, −4.42524768227606600015483558127, −4.38165575799717027504601939918, −4.32027392561376677001745414363, −3.94311454616842355303373312775, −3.85690095372589472487771417985, −3.84604797838958655183972646791, −3.30379945839716216775303106710, −3.09693223126451561764672700549, −3.06665297025647776543043689690, −3.00561425281248799592997466087, −2.50985408912471776475452362162, −2.14828866343127166663724960068, −2.02166198656269247126603577261, −1.94839501395435251777662814835, −1.86455690575313592563844226168, −1.78566732247443588297862849371, −0.66976178330340368243479010491, −0.63449745383716630292278610059, −0.58364496941621327173394633945,
0.58364496941621327173394633945, 0.63449745383716630292278610059, 0.66976178330340368243479010491, 1.78566732247443588297862849371, 1.86455690575313592563844226168, 1.94839501395435251777662814835, 2.02166198656269247126603577261, 2.14828866343127166663724960068, 2.50985408912471776475452362162, 3.00561425281248799592997466087, 3.06665297025647776543043689690, 3.09693223126451561764672700549, 3.30379945839716216775303106710, 3.84604797838958655183972646791, 3.85690095372589472487771417985, 3.94311454616842355303373312775, 4.32027392561376677001745414363, 4.38165575799717027504601939918, 4.42524768227606600015483558127, 4.61482243608506239092839840531, 4.69839459516157866001107771637, 5.07795171912515974949868654564, 5.28125687756670293508646874366, 5.31865282800980099572711364246, 5.59608336081598638679179934646
Plot not available for L-functions of degree greater than 10.
