| L(s) = 1 | − 6·7-s + 4·9-s − 4·17-s + 8·23-s − 3·25-s + 16·31-s − 36·41-s − 20·47-s + 21·49-s − 24·63-s + 48·71-s − 44·73-s − 16·79-s − 52·89-s + 28·97-s − 52·103-s − 4·113-s + 24·119-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 4/3·9-s − 0.970·17-s + 1.66·23-s − 3/5·25-s + 2.87·31-s − 5.62·41-s − 2.91·47-s + 3·49-s − 3.02·63-s + 5.69·71-s − 5.14·73-s − 1.80·79-s − 5.51·89-s + 2.84·97-s − 5.12·103-s − 0.376·113-s + 2.20·119-s + 2.54·121-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 − 1.29·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.073282538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.073282538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 7 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - 4 T^{2} + 16 T^{4} - 62 T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 28 T^{2} + 344 T^{4} - 3354 T^{6} + 344 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 32 T^{2} + 64 p T^{4} - 11846 T^{6} + 64 p^{3} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T - 4 T^{2} - 38 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 58 T^{2} + 1943 T^{4} - 44652 T^{6} + 1943 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 9 T^{2} - 24 T^{3} + 9 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 72 T^{2} + 4136 T^{4} - 132218 T^{6} + 4136 p^{2} T^{8} - 72 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 8 T + 69 T^{2} - 464 T^{3} + 69 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 54 T^{2} + 167 T^{4} + 59788 T^{6} + 167 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 18 T + 203 T^{2} + 1556 T^{3} + 203 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{3} \) |
| 47 | \( ( 1 + 10 T + 154 T^{2} + 876 T^{3} + 154 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 142 T^{2} + 11543 T^{4} - 697956 T^{6} + 11543 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 178 T^{2} + 17399 T^{4} - 1199004 T^{6} + 17399 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 74 T^{2} + 823 T^{4} + 21940 T^{6} + 823 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 238 T^{2} + 29207 T^{4} - 2378244 T^{6} + 29207 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 24 T + 377 T^{2} - 3728 T^{3} + 377 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 22 T + 315 T^{2} + 3052 T^{3} + 315 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 8 T + 58 T^{2} - 208 T^{3} + 58 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 442 T^{2} + 85527 T^{4} - 9250156 T^{6} + 85527 p^{2} T^{8} - 442 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 26 T + 427 T^{2} + 4788 T^{3} + 427 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 14 T + 300 T^{2} - 2486 T^{3} + 300 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.71393121014216437546103253741, −4.58083827563206496620632127968, −4.46581491408637516785406065863, −4.19014763308781626847662267030, −3.97812349003314518335649740095, −3.95453118515219886718180023293, −3.80156278155615340517773754188, −3.72490319127573706161840418341, −3.35969732343852794289586471881, −3.27688119723841179328246677484, −3.17571515127938599391635393301, −2.93491750803413999228462643034, −2.83285097433587061983130582085, −2.73748813522649728506353117965, −2.56502039389646904231701807476, −2.51662502219892499514955126198, −1.94655996479284986221289336963, −1.83237015821754858248308187473, −1.61587753810156607365718105453, −1.48640271325058404635576051631, −1.37355136144138289882015142686, −1.22276126376373067973429259391, −0.56563651343461645109214901649, −0.40992419445077651795444675953, −0.28030674977642469518315792500,
0.28030674977642469518315792500, 0.40992419445077651795444675953, 0.56563651343461645109214901649, 1.22276126376373067973429259391, 1.37355136144138289882015142686, 1.48640271325058404635576051631, 1.61587753810156607365718105453, 1.83237015821754858248308187473, 1.94655996479284986221289336963, 2.51662502219892499514955126198, 2.56502039389646904231701807476, 2.73748813522649728506353117965, 2.83285097433587061983130582085, 2.93491750803413999228462643034, 3.17571515127938599391635393301, 3.27688119723841179328246677484, 3.35969732343852794289586471881, 3.72490319127573706161840418341, 3.80156278155615340517773754188, 3.95453118515219886718180023293, 3.97812349003314518335649740095, 4.19014763308781626847662267030, 4.46581491408637516785406065863, 4.58083827563206496620632127968, 4.71393121014216437546103253741
Plot not available for L-functions of degree greater than 10.
