| L(s) = 1 | + 6·19-s + 6·37-s − 11·64-s − 12·73-s + 204·109-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 + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 1.37·19-s + 0.986·37-s − 1.37·64-s − 1.40·73-s + 19.5·109-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.200627597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.200627597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 11 T^{6} + 57 T^{12} + 11 p^{6} T^{18} + p^{12} T^{24} \) |
| 5 | \( 1 - 142 T^{6} + 4539 T^{12} - 142 p^{6} T^{18} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 17 T^{3} + p^{3} T^{6} )^{2}( 1 + 37 T^{3} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 + 2630 T^{6} + 5145339 T^{12} + 2630 p^{6} T^{18} + p^{12} T^{24} \) |
| 13 | \( ( 1 - 89 T^{3} + p^{3} T^{6} )^{2}( 1 + 19 T^{3} + p^{3} T^{6} )^{2} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{6} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{6}( 1 + 7 T + p T^{2} )^{6} \) |
| 23 | \( 1 - 3166 T^{6} - 138012333 T^{12} - 3166 p^{6} T^{18} + p^{12} T^{24} \) |
| 29 | \( 1 + 18722 T^{6} - 244310037 T^{12} + 18722 p^{6} T^{18} + p^{12} T^{24} \) |
| 31 | \( ( 1 - 340 T^{3} + 85809 T^{6} - 340 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 11 T + p T^{2} )^{6}( 1 + 10 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 10010 T^{6} - 4649904141 T^{12} + 10010 p^{6} T^{18} + p^{12} T^{24} \) |
| 43 | \( ( 1 + 128 T^{3} - 63123 T^{6} + 128 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 - 7954 T^{6} - 10715949213 T^{12} - 7954 p^{6} T^{18} + p^{12} T^{24} \) |
| 53 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{6} \) |
| 59 | \( 1 - 84058 T^{6} - 35114786277 T^{12} - 84058 p^{6} T^{18} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 358 T^{3} - 98817 T^{6} - 358 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 1096 T^{3} + 900453 T^{6} - 1096 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 34 T^{2} - 3885 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 73 | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{6} \) |
| 79 | \( ( 1 + 236 T^{3} - 437343 T^{6} + 236 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 795674 T^{6} + 306156740907 T^{12} + 795674 p^{6} T^{18} + p^{12} T^{24} \) |
| 89 | \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 97 | \( ( 1 - 34 T^{3} - 911517 T^{6} - 34 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.30676864628407184998818128959, −3.24614602853304289953305633151, −3.23168591085208729326122769740, −3.22466258313304849425763074670, −3.19886469357307466710158690200, −3.03874877494723370325247381901, −2.69807831960993803829691003113, −2.55624458821947458889557806415, −2.42420870351103210421677174294, −2.41784062240076514268593010533, −2.41283335485612933447174831786, −2.21918717256222236748963426204, −2.06873494140408445931418024130, −2.04163807764496035973278188877, −2.02264235311953138203832965823, −1.60532324430562213869251787458, −1.60089700235212566798291976901, −1.44927442163203156253429363928, −1.31639878977277672506223145894, −1.19475357274974093344576100748, −0.921340825796910299897261123653, −0.879358233989706670386148057197, −0.69099897615432776937132617573, −0.46999858994274609726907188939, −0.16261469320101544249087164867,
0.16261469320101544249087164867, 0.46999858994274609726907188939, 0.69099897615432776937132617573, 0.879358233989706670386148057197, 0.921340825796910299897261123653, 1.19475357274974093344576100748, 1.31639878977277672506223145894, 1.44927442163203156253429363928, 1.60089700235212566798291976901, 1.60532324430562213869251787458, 2.02264235311953138203832965823, 2.04163807764496035973278188877, 2.06873494140408445931418024130, 2.21918717256222236748963426204, 2.41283335485612933447174831786, 2.41784062240076514268593010533, 2.42420870351103210421677174294, 2.55624458821947458889557806415, 2.69807831960993803829691003113, 3.03874877494723370325247381901, 3.19886469357307466710158690200, 3.22466258313304849425763074670, 3.23168591085208729326122769740, 3.24614602853304289953305633151, 3.30676864628407184998818128959
Plot not available for L-functions of degree greater than 10.
