L(s) = 1 | + 4·9-s − 2·11-s − 6·19-s − 6·29-s − 2·31-s − 6·41-s + 12·59-s − 10·61-s − 6·71-s − 4·79-s − 12·81-s − 28·89-s − 8·99-s + 8·101-s − 10·109-s − 51·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 57·169-s − 24·171-s + ⋯ |
L(s) = 1 | + 4/3·9-s − 0.603·11-s − 1.37·19-s − 1.11·29-s − 0.359·31-s − 0.937·41-s + 1.56·59-s − 1.28·61-s − 0.712·71-s − 0.450·79-s − 4/3·81-s − 2.96·89-s − 0.804·99-s + 0.796·101-s − 0.957·109-s − 4.63·121-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 + 4.38·169-s − 1.83·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.206769294\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.206769294\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
good | 3 | \( 1 - 4 T^{2} + 28 T^{4} - 23 p T^{6} + 28 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 108 T^{4} + 11 T^{6} + 108 p^{2} T^{8} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 57 T^{2} + 9 p^{2} T^{4} - 24625 T^{6} + 9 p^{4} T^{8} - 57 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 36 T^{2} + 1260 T^{4} - 22057 T^{6} + 1260 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 44 T^{2} + 1720 T^{4} - 40261 T^{6} + 1720 p^{2} T^{8} - 44 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 3 T + 63 T^{2} + 201 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + T + 53 T^{2} - 47 T^{3} + 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 153 T^{2} + 11289 T^{4} - 514297 T^{6} + 11289 p^{2} T^{8} - 153 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 3 T + 87 T^{2} + 219 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 151 T^{2} + 12086 T^{4} - 635251 T^{6} + 12086 p^{2} T^{8} - 151 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 9 T^{2} + 2205 T^{4} + 10487 T^{6} + 2205 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 179 T^{2} + 17422 T^{4} - 1138855 T^{6} + 17422 p^{2} T^{8} - 179 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 6 T + 153 T^{2} - 636 T^{3} + 153 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 5 T + 85 T^{2} + 121 T^{3} + 85 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 11 T^{2} + 7601 T^{4} + 31511 T^{6} + 7601 p^{2} T^{8} + 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 3 T + 180 T^{2} + 399 T^{3} + 180 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 297 T^{2} + 41997 T^{4} - 3741769 T^{6} + 41997 p^{2} T^{8} - 297 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 2 T + 80 T^{2} + 845 T^{3} + 80 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 209 T^{2} + 33625 T^{4} - 3104401 T^{6} + 33625 p^{2} T^{8} - 209 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 14 T + 258 T^{2} + 2003 T^{3} + 258 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 273 T^{2} + 43593 T^{4} - 4678585 T^{6} + 43593 p^{2} T^{8} - 273 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.76088168281691657954980001989, −4.75896372629434428057029398859, −4.56618596651546593130703658672, −4.25921869521416877917222401275, −4.06802917004347539563326385739, −4.01053282713114565255934105742, −3.99713760387415920214192370736, −3.74850398946830646678724799502, −3.68159555660128520692386505775, −3.60034443888112618453715339169, −3.11652296225793016088539528711, −2.92268701565442957143380155658, −2.80535686771350089356006000795, −2.79777837051996124359398534722, −2.59053216591866778427964775256, −2.48195497936928768087427527540, −2.02369863610681316443641478784, −1.87009815650227618016818748658, −1.72974327089911448715816497667, −1.47574584357724312042331297687, −1.45901296975055003881930504946, −1.38207456248595194944674107436, −0.61385105738458630545453482370, −0.45074548659929498168965681507, −0.37524737599855903835700058134,
0.37524737599855903835700058134, 0.45074548659929498168965681507, 0.61385105738458630545453482370, 1.38207456248595194944674107436, 1.45901296975055003881930504946, 1.47574584357724312042331297687, 1.72974327089911448715816497667, 1.87009815650227618016818748658, 2.02369863610681316443641478784, 2.48195497936928768087427527540, 2.59053216591866778427964775256, 2.79777837051996124359398534722, 2.80535686771350089356006000795, 2.92268701565442957143380155658, 3.11652296225793016088539528711, 3.60034443888112618453715339169, 3.68159555660128520692386505775, 3.74850398946830646678724799502, 3.99713760387415920214192370736, 4.01053282713114565255934105742, 4.06802917004347539563326385739, 4.25921869521416877917222401275, 4.56618596651546593130703658672, 4.75896372629434428057029398859, 4.76088168281691657954980001989
Plot not available for L-functions of degree greater than 10.