| L(s) = 1 | + 12·5-s + 10·9-s + 16-s + 78·25-s + 4·37-s + 120·45-s − 46·49-s − 4·53-s + 4·64-s + 12·80-s + 33·81-s + 36·89-s − 8·97-s + 24·113-s − 10·121-s + 364·125-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯ |
| L(s) = 1 | + 5.36·5-s + 10/3·9-s + 1/4·16-s + 78/5·25-s + 0.657·37-s + 17.8·45-s − 6.57·49-s − 0.549·53-s + 1/2·64-s + 1.34·80-s + 11/3·81-s + 3.81·89-s − 0.812·97-s + 2.25·113-s − 0.909·121-s + 32.5·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.52464371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.52464371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} - p^{2} T^{6} - p^{2} T^{8} + p^{6} T^{12} \) |
| 5 | \( ( 1 - T )^{12} \) |
| 11 | \( 1 + 10 T^{2} + 71 T^{4} + 1548 T^{6} + 71 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 3 | \( ( 1 - 5 T^{2} + 7 p T^{4} - 82 T^{6} + 7 p^{3} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 + 23 T^{2} + 271 T^{4} + 2162 T^{6} + 271 p^{2} T^{8} + 23 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 36 T^{2} + 887 T^{4} - 13240 T^{6} + 887 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 63 T^{2} + 1769 T^{4} - 33526 T^{6} + 1769 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 5 p T^{2} + 4039 T^{4} + 98546 T^{6} + 4039 p^{2} T^{8} + 5 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 108 T^{2} + 5423 T^{4} - 159016 T^{6} + 5423 p^{2} T^{8} - 108 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 77 T^{2} + 3035 T^{4} - 89022 T^{6} + 3035 p^{2} T^{8} - 77 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 135 T^{2} + 8735 T^{4} - 341554 T^{6} + 8735 p^{2} T^{8} - 135 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - T + 31 T^{2} - 286 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 70 T^{2} + 3007 T^{4} - 116308 T^{6} + 3007 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 118 T^{2} + 6727 T^{4} + 284788 T^{6} + 6727 p^{2} T^{8} + 118 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 12 T^{2} + 2463 T^{4} - 12904 T^{6} + 2463 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + T + 107 T^{2} + 230 T^{3} + 107 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( ( 1 - 270 T^{2} + 32807 T^{4} - 2401732 T^{6} + 32807 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 301 T^{2} + 41283 T^{4} - 3242254 T^{6} + 41283 p^{2} T^{8} - 301 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 104 T^{2} + 9935 T^{4} - 659616 T^{6} + 9935 p^{2} T^{8} - 104 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 239 T^{2} + 30959 T^{4} - 2696130 T^{6} + 30959 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 100 T^{2} + 13455 T^{4} + 959608 T^{6} + 13455 p^{2} T^{8} + 100 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 146 T^{2} + 20831 T^{4} - 1830108 T^{6} + 20831 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 234 T^{2} + 29975 T^{4} + 2694668 T^{6} + 29975 p^{2} T^{8} + 234 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 9 T + 165 T^{2} - 938 T^{3} + 165 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 + 2 T - 33 T^{2} - 4 T^{3} - 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.27632374393016544779193315144, −4.11011158411645853064363928512, −4.06603420002457643688347955046, −4.05757895536378426861978937982, −3.87048125769078804390204009766, −3.75754033695638333511766303319, −3.63202919247051812961959745631, −3.26514572116430547240248648191, −3.25144682741088494546602358675, −3.12752877759680587859872222956, −2.99201337623790660238674931330, −2.94324033466690018803420983697, −2.88393691080233970314463087343, −2.46491672664236253472528501560, −2.33192419550725962867296083877, −2.30761548835383446729383481953, −2.28232913634462863517935612389, −1.90912497607685358913359579227, −1.74571274180383667961085625767, −1.71844605583125438996747489063, −1.69148623158413078306913117523, −1.34539865300933431253980709046, −1.30298746480378800197004827864, −1.25823738211269500436308085280, −0.808740271005341622398044069913,
0.808740271005341622398044069913, 1.25823738211269500436308085280, 1.30298746480378800197004827864, 1.34539865300933431253980709046, 1.69148623158413078306913117523, 1.71844605583125438996747489063, 1.74571274180383667961085625767, 1.90912497607685358913359579227, 2.28232913634462863517935612389, 2.30761548835383446729383481953, 2.33192419550725962867296083877, 2.46491672664236253472528501560, 2.88393691080233970314463087343, 2.94324033466690018803420983697, 2.99201337623790660238674931330, 3.12752877759680587859872222956, 3.25144682741088494546602358675, 3.26514572116430547240248648191, 3.63202919247051812961959745631, 3.75754033695638333511766303319, 3.87048125769078804390204009766, 4.05757895536378426861978937982, 4.06603420002457643688347955046, 4.11011158411645853064363928512, 4.27632374393016544779193315144
Plot not available for L-functions of degree greater than 10.
