| L(s) = 1 | + 6·3-s + 21·9-s + 56·27-s + 12·31-s − 8·37-s − 20·41-s − 8·43-s + 6·49-s − 12·53-s + 24·67-s + 8·71-s + 36·79-s + 126·81-s + 32·83-s + 28·89-s + 72·93-s + 40·107-s − 48·111-s + 2·121-s − 120·123-s + 127-s − 48·129-s + 131-s + 137-s + 139-s + 36·147-s + 149-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 7·9-s + 10.7·27-s + 2.15·31-s − 1.31·37-s − 3.12·41-s − 1.21·43-s + 6/7·49-s − 1.64·53-s + 2.93·67-s + 0.949·71-s + 4.05·79-s + 14·81-s + 3.51·83-s + 2.96·89-s + 7.46·93-s + 3.86·107-s − 4.55·111-s + 2/11·121-s − 10.8·123-s + 0.0887·127-s − 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.96·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(40.27011101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(40.27011101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 11 T^{2} - 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 4 T + 51 T^{2} + 40 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 4 T + 65 T^{2} + 216 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{6} \) |
| 71 | \( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.75548612646811221649308172778, −4.56414167715133082915611853668, −4.23828884846416023973063054672, −4.00636078270904823089089889224, −3.95006113287529293088429752837, −3.68607614862327999040566169900, −3.58914634324196703489370340258, −3.58259352866861253422616305858, −3.39856518645524919302969834009, −3.32996823022288471911372854979, −3.29770649476703901681287696962, −2.96919658027631263968802931052, −2.78850816464886513641144076088, −2.49583725700872458240328398898, −2.47695400933385392709423147985, −2.17524481150253556806387924748, −2.16922650646852252975210900009, −2.05236991245424410707104141295, −1.86004755616638252934571693435, −1.64652685770277551560491194768, −1.36528714491591577329387285995, −1.20203360544240795621066385997, −0.75956929801200665649749317828, −0.75908279633423591072970556629, −0.37674848081668442111708560378,
0.37674848081668442111708560378, 0.75908279633423591072970556629, 0.75956929801200665649749317828, 1.20203360544240795621066385997, 1.36528714491591577329387285995, 1.64652685770277551560491194768, 1.86004755616638252934571693435, 2.05236991245424410707104141295, 2.16922650646852252975210900009, 2.17524481150253556806387924748, 2.47695400933385392709423147985, 2.49583725700872458240328398898, 2.78850816464886513641144076088, 2.96919658027631263968802931052, 3.29770649476703901681287696962, 3.32996823022288471911372854979, 3.39856518645524919302969834009, 3.58259352866861253422616305858, 3.58914634324196703489370340258, 3.68607614862327999040566169900, 3.95006113287529293088429752837, 4.00636078270904823089089889224, 4.23828884846416023973063054672, 4.56414167715133082915611853668, 4.75548612646811221649308172778
Plot not available for L-functions of degree greater than 10.
