L(s) = 1 | − 2·2-s + 4-s − 7-s − 9-s − 2·11-s + 2·14-s + 2·18-s + 4·22-s − 2·23-s − 25-s − 28-s − 2·29-s − 36-s − 2·37-s + 9·43-s − 2·44-s + 4·46-s + 2·50-s − 2·53-s + 4·58-s + 63-s − 67-s − 2·71-s + 4·74-s + 2·77-s − 2·79-s − 18·86-s + ⋯ |
L(s) = 1 | − 2·2-s + 4-s − 7-s − 9-s − 2·11-s + 2·14-s + 2·18-s + 4·22-s − 2·23-s − 25-s − 28-s − 2·29-s − 36-s − 2·37-s + 9·43-s − 2·44-s + 4·46-s + 2·50-s − 2·53-s + 4·58-s + 63-s − 67-s − 2·71-s + 4·74-s + 2·77-s − 2·79-s − 18·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01620698384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01620698384\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 67 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
good | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 11 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 37 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 43 | \( ( 1 - T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 53 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 79 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 97 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.44912739451439535973745617642, −4.43268749083018766211002302452, −4.34338127450241957603406942601, −3.91727717231766457771980448609, −3.78750047796153468666379660192, −3.77877087064149854789193967602, −3.74000206505801959231416453856, −3.72123818755198054483908870701, −3.55569260471196365530993563617, −3.31271276417703868900813254140, −3.18506514476697953806199191554, −3.00549345907325947536084911028, −2.78321963887218509200221216408, −2.64832819021255328189491533569, −2.59000915139312289234792076365, −2.56330671936519172331505049897, −2.53421684425009053283185214436, −2.30748536120877551085496046129, −1.90578125948631098476772234133, −1.78834633499585478704300734494, −1.76859206316616135951409823355, −1.72667752669369234450281766650, −1.15481487553490089905230847371, −0.967089883061366918153781760922, −0.43807553727538239498008981235,
0.43807553727538239498008981235, 0.967089883061366918153781760922, 1.15481487553490089905230847371, 1.72667752669369234450281766650, 1.76859206316616135951409823355, 1.78834633499585478704300734494, 1.90578125948631098476772234133, 2.30748536120877551085496046129, 2.53421684425009053283185214436, 2.56330671936519172331505049897, 2.59000915139312289234792076365, 2.64832819021255328189491533569, 2.78321963887218509200221216408, 3.00549345907325947536084911028, 3.18506514476697953806199191554, 3.31271276417703868900813254140, 3.55569260471196365530993563617, 3.72123818755198054483908870701, 3.74000206505801959231416453856, 3.77877087064149854789193967602, 3.78750047796153468666379660192, 3.91727717231766457771980448609, 4.34338127450241957603406942601, 4.43268749083018766211002302452, 4.44912739451439535973745617642
Plot not available for L-functions of degree greater than 10.