L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 9-s − 12-s + 2·13-s − 17-s + 18-s − 19-s + 23-s − 3·25-s + 2·26-s − 34-s + 36-s + 37-s − 38-s − 2·39-s − 6·41-s − 2·43-s + 46-s − 47-s − 3·50-s + 51-s + 2·52-s + 57-s − 68-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 9-s − 12-s + 2·13-s − 17-s + 18-s − 19-s + 23-s − 3·25-s + 2·26-s − 34-s + 36-s + 37-s − 38-s − 2·39-s − 6·41-s − 2·43-s + 46-s − 47-s − 3·50-s + 51-s + 2·52-s + 57-s − 68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3580768895\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3580768895\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( ( 1 + T )^{6} \) |
good | 2 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 3 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 5 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 17 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 19 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 23 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 29 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 31 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 37 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 43 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 47 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 53 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 59 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 61 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 67 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 79 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 83 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 89 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
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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
−5.04401577427435506798130622400, −4.82836953762482163783993782437, −4.63375378762089689495879156420, −4.55516777346898850179205652688, −4.50092349371456262817860635579, −4.24639969694681809920604621951, −3.93556652893087677963129953892, −3.90542753497502932814445841013, −3.87345243392989194229338668329, −3.60603968135986095328287220566, −3.54652072770348459017211618959, −3.24868771091844489836162223007, −3.15988414154023773290477849106, −3.07571694836446014133026789368, −2.96112448191744686442098279712, −2.53557174314562538911207541897, −2.38895967264862872838212690630, −2.13557826763703809556282800369, −2.03449118281866938440195291613, −1.76580778284001710353102003885, −1.65919010299767152506857707676, −1.39791643264655361206048294006, −1.28325315793794628384740804361, −1.03645408969807067672409411965, −0.20020009897710312996831977629,
0.20020009897710312996831977629, 1.03645408969807067672409411965, 1.28325315793794628384740804361, 1.39791643264655361206048294006, 1.65919010299767152506857707676, 1.76580778284001710353102003885, 2.03449118281866938440195291613, 2.13557826763703809556282800369, 2.38895967264862872838212690630, 2.53557174314562538911207541897, 2.96112448191744686442098279712, 3.07571694836446014133026789368, 3.15988414154023773290477849106, 3.24868771091844489836162223007, 3.54652072770348459017211618959, 3.60603968135986095328287220566, 3.87345243392989194229338668329, 3.90542753497502932814445841013, 3.93556652893087677963129953892, 4.24639969694681809920604621951, 4.50092349371456262817860635579, 4.55516777346898850179205652688, 4.63375378762089689495879156420, 4.82836953762482163783993782437, 5.04401577427435506798130622400
Plot not available for L-functions of degree greater than 10.