| L(s) = 1 | − 2-s + 4-s − 8-s + 8·13-s + 16-s + 25-s − 8·26-s − 16·31-s − 5·32-s + 16·37-s + 4·41-s + 18·49-s − 50-s + 8·52-s + 24·53-s + 16·62-s + 64-s − 16·71-s − 16·74-s + 16·79-s − 4·82-s − 16·83-s + 20·89-s − 18·98-s + 100-s − 8·104-s − 24·106-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2.21·13-s + 1/4·16-s + 1/5·25-s − 1.56·26-s − 2.87·31-s − 0.883·32-s + 2.63·37-s + 0.624·41-s + 18/7·49-s − 0.141·50-s + 1.10·52-s + 3.29·53-s + 2.03·62-s + 1/8·64-s − 1.89·71-s − 1.85·74-s + 1.80·79-s − 0.441·82-s − 1.75·83-s + 2.11·89-s − 1.81·98-s + 1/10·100-s − 0.784·104-s − 2.33·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132633625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132633625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T^{2} - 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 4 T + 23 T^{2} - 48 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 2 T + 23 T^{2} - 220 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 12 T + 191 T^{2} - 1264 T^{3} + 191 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 137 T^{2} + 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 8 T + 185 T^{2} + 880 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−6.10015587382555690239644792156, −6.07489428590315759746169442752, −5.96811472810705703859222832841, −5.87461014716662849957506604185, −5.60811020835615046518081378886, −5.35936636704649422525182040043, −5.18046351619967547590411239353, −4.97263153939161633294388923312, −4.78502043571790355248690623310, −4.50582818968868191961432733130, −4.11484246978006869203802557296, −3.96076908310710617481020453079, −3.90790894059386401818841036155, −3.88992291072403147187779700442, −3.53152812415233582771992761570, −3.24594377001841507187673508350, −3.06204124173003800230703701499, −2.68714512932063640124658303348, −2.48315439361145514890695933474, −2.17787959758663038483908588786, −2.09825786602457096527844522366, −1.60176374548220699099568604134, −1.22089145497875338021484942545, −1.03609889396730616685065189693, −0.54231708773325161022118296356,
0.54231708773325161022118296356, 1.03609889396730616685065189693, 1.22089145497875338021484942545, 1.60176374548220699099568604134, 2.09825786602457096527844522366, 2.17787959758663038483908588786, 2.48315439361145514890695933474, 2.68714512932063640124658303348, 3.06204124173003800230703701499, 3.24594377001841507187673508350, 3.53152812415233582771992761570, 3.88992291072403147187779700442, 3.90790894059386401818841036155, 3.96076908310710617481020453079, 4.11484246978006869203802557296, 4.50582818968868191961432733130, 4.78502043571790355248690623310, 4.97263153939161633294388923312, 5.18046351619967547590411239353, 5.35936636704649422525182040043, 5.60811020835615046518081378886, 5.87461014716662849957506604185, 5.96811472810705703859222832841, 6.07489428590315759746169442752, 6.10015587382555690239644792156
Plot not available for L-functions of degree greater than 10.
