L(s) = 1 | + 8-s + 3·41-s − 3·49-s + 3·59-s − 3·67-s + 6·73-s − 3·97-s − 3·107-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 + ⋯ |
L(s) = 1 | + 8-s + 3·41-s − 3·49-s + 3·59-s − 3·67-s + 6·73-s − 3·97-s − 3·107-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.206099994\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.206099994\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T^{3} + T^{6} \) |
good | 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 11 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 13 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 17 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 23 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 29 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 31 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 37 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 41 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 43 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 47 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 53 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 59 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 61 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 71 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 73 | \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \) |
| 79 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 83 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 89 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 97 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
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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
−5.34917796674674607365311047076, −5.10322673611250113940444738004, −4.87527718899260544314940461264, −4.82474859965014723984427307759, −4.60591103935087721741175043268, −4.60120072680533270478644518944, −4.32776739984497326455100462331, −4.06014770763908497093717848947, −4.02334268265414628921121604209, −3.75770325074225807551660191206, −3.70676610531026564349526626369, −3.59426921854885338489579199497, −3.51170385586158802213469212714, −3.00082475588478984637833464627, −2.86846714021961662671051209902, −2.71659461309361275623900224654, −2.52076218624501440137204669672, −2.39820065182450849368983229293, −2.38080897339282886204784830074, −1.89424456873575858825120713011, −1.58900805298940183490429102612, −1.47620533384632479922595983506, −1.44625138924848356055359883944, −0.936260874972603277636415509724, −0.69525114263984237989861565491,
0.69525114263984237989861565491, 0.936260874972603277636415509724, 1.44625138924848356055359883944, 1.47620533384632479922595983506, 1.58900805298940183490429102612, 1.89424456873575858825120713011, 2.38080897339282886204784830074, 2.39820065182450849368983229293, 2.52076218624501440137204669672, 2.71659461309361275623900224654, 2.86846714021961662671051209902, 3.00082475588478984637833464627, 3.51170385586158802213469212714, 3.59426921854885338489579199497, 3.70676610531026564349526626369, 3.75770325074225807551660191206, 4.02334268265414628921121604209, 4.06014770763908497093717848947, 4.32776739984497326455100462331, 4.60120072680533270478644518944, 4.60591103935087721741175043268, 4.82474859965014723984427307759, 4.87527718899260544314940461264, 5.10322673611250113940444738004, 5.34917796674674607365311047076
Plot not available for L-functions of degree greater than 10.