L(s) = 1 | − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s − 9·79-s + 45·81-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 + ⋯ |
L(s) = 1 | − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s − 9·79-s + 45·81-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1999^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1999^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.017735772\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.017735772\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1999 | \( ( 1 - T )^{9} \) |
good | 2 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
| 3 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 5 | \( 1 + T^{9} + T^{18} \) |
| 7 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 11 | \( 1 + T^{9} + T^{18} \) |
| 13 | \( 1 + T^{9} + T^{18} \) |
| 17 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 19 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 23 | \( 1 + T^{9} + T^{18} \) |
| 29 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 31 | \( 1 + T^{9} + T^{18} \) |
| 37 | \( 1 + T^{9} + T^{18} \) |
| 41 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
| 43 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 47 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 53 | \( 1 + T^{9} + T^{18} \) |
| 59 | \( 1 + T^{9} + T^{18} \) |
| 61 | \( 1 + T^{9} + T^{18} \) |
| 67 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 71 | \( 1 + T^{9} + T^{18} \) |
| 73 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 79 | \( ( 1 + T + T^{2} )^{9} \) |
| 83 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 89 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 97 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.79176762952031764320270421991, −3.75399302478977961172276974199, −3.67011388962456349578618538585, −3.62941793363368733333421194876, −3.59300015773679058159407408618, −3.20423107783851481412107319081, −2.92336681090380113213081444336, −2.90558538318245228780441084833, −2.85763883781879452377108217141, −2.80772190251028919032264138316, −2.48971949477049115945887030453, −2.44973823615711600982815263934, −2.39422333146600656434440557447, −2.30643809889651343823578983846, −2.05411147918019502541854764261, −1.89838391853538569543298357597, −1.86448843237071398693566789467, −1.63498344347997472894935154832, −1.41156201859132925349198587018, −1.35895581883200150744010758821, −1.30659825116056007744396350025, −1.25276434109426018947382419609, −0.960235859488580258087108771962, −0.799026337350778006674919180746, −0.77478281085216062984192416276,
0.77478281085216062984192416276, 0.799026337350778006674919180746, 0.960235859488580258087108771962, 1.25276434109426018947382419609, 1.30659825116056007744396350025, 1.35895581883200150744010758821, 1.41156201859132925349198587018, 1.63498344347997472894935154832, 1.86448843237071398693566789467, 1.89838391853538569543298357597, 2.05411147918019502541854764261, 2.30643809889651343823578983846, 2.39422333146600656434440557447, 2.44973823615711600982815263934, 2.48971949477049115945887030453, 2.80772190251028919032264138316, 2.85763883781879452377108217141, 2.90558538318245228780441084833, 2.92336681090380113213081444336, 3.20423107783851481412107319081, 3.59300015773679058159407408618, 3.62941793363368733333421194876, 3.67011388962456349578618538585, 3.75399302478977961172276974199, 3.79176762952031764320270421991
Plot not available for L-functions of degree greater than 10.