| L(s) = 1 | − 49·7-s + 175·25-s + 287·41-s + 1.37e3·49-s + 847·121-s + 127-s − 31·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 8.57e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 7·7-s + 7·25-s + 7·41-s + 28·49-s + 7·121-s + 0.00787·127-s − 0.242·128-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s − 49·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{7} \cdot 41^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{7} \cdot 41^{7}\right)^{s/2} \, \Gamma_{\C}(s+1)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.903669697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.903669697\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + p T )^{7} \) |
| 41 | \( ( 1 - p T )^{7} \) |
| good | 2 | \( 1 + 31 T^{7} + p^{14} T^{14} \) |
| 3 | \( 1 + 3718 T^{7} + p^{14} T^{14} \) |
| 5 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 11 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 13 | \( 1 + 24950438 T^{7} + p^{14} T^{14} \) |
| 17 | \( 1 + 800009246 T^{7} + p^{14} T^{14} \) |
| 19 | \( 1 + 305833574 T^{7} + p^{14} T^{14} \) |
| 23 | \( 1 - 6483516722 T^{7} + p^{14} T^{14} \) |
| 29 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 31 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 37 | \( 1 - 101889625898 T^{7} + p^{14} T^{14} \) |
| 43 | \( 1 - 358855177802 T^{7} + p^{14} T^{14} \) |
| 47 | \( 1 - 987930187618 T^{7} + p^{14} T^{14} \) |
| 53 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 59 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 61 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 67 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 71 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 73 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 79 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 83 | \( ( 1 - p T )^{7}( 1 + p T )^{7} \) |
| 89 | \( 1 - 87666289649746 T^{7} + p^{14} T^{14} \) |
| 97 | \( 1 - 74677982531458 T^{7} + p^{14} T^{14} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65672611929260821359838087879, −5.60077882017353762638357039311, −5.35667300140548979292496920336, −5.16592730490314960806696617661, −4.78019594529829108896004563810, −4.58116732589004632666082809185, −4.51556712705199215339340489020, −4.16059233239379040205061914312, −4.13465189795418057223549854897, −4.10207606168342537845641808861, −3.57068360733795824981449457470, −3.44833566716796602454560871823, −3.29425159204719963136178765038, −3.19822775608545646370322877835, −3.01955250338513874442996835387, −2.90935211523299742141256627500, −2.69269007198505005667560617691, −2.47815288581083905408668072746, −2.44049277061994159604840064260, −2.27525055906270898754169847478, −1.26334414844324904248444218967, −0.879332753478647521789236891062, −0.76828959163404823861685214935, −0.57778164861474880804161102853, −0.48052501210901266852092307548,
0.48052501210901266852092307548, 0.57778164861474880804161102853, 0.76828959163404823861685214935, 0.879332753478647521789236891062, 1.26334414844324904248444218967, 2.27525055906270898754169847478, 2.44049277061994159604840064260, 2.47815288581083905408668072746, 2.69269007198505005667560617691, 2.90935211523299742141256627500, 3.01955250338513874442996835387, 3.19822775608545646370322877835, 3.29425159204719963136178765038, 3.44833566716796602454560871823, 3.57068360733795824981449457470, 4.10207606168342537845641808861, 4.13465189795418057223549854897, 4.16059233239379040205061914312, 4.51556712705199215339340489020, 4.58116732589004632666082809185, 4.78019594529829108896004563810, 5.16592730490314960806696617661, 5.35667300140548979292496920336, 5.60077882017353762638357039311, 5.65672611929260821359838087879
Plot not available for L-functions of degree greater than 10.
