| L(s) = 1 | + 5·9-s + 14·11-s + 8·19-s − 5·25-s − 6·29-s − 20·31-s − 36·41-s + 12·59-s − 48·61-s + 8·71-s − 34·79-s + 14·81-s + 70·99-s + 8·101-s − 10·109-s + 65·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 4.22·11-s + 1.83·19-s − 25-s − 1.11·29-s − 3.59·31-s − 5.62·41-s + 1.56·59-s − 6.14·61-s + 0.949·71-s − 3.82·79-s + 14/9·81-s + 7.03·99-s + 0.796·101-s − 0.957·109-s + 5.90·121-s + 0.715·125-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 + 9/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.574952565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.574952565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T^{2} - 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 5 T^{2} + 11 T^{4} - 26 T^{6} + 11 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 7 T + 41 T^{2} - 146 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 491 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 1539 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 4 T + 43 T^{2} - 160 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 2 p T^{2} + 1887 T^{4} - 43972 T^{6} + 1887 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 3 T + 15 T^{2} + 66 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 10 T + 101 T^{2} + 540 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{3} \) |
| 41 | \( ( 1 + 18 T + 191 T^{2} + 1388 T^{3} + 191 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 15575 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 6339 p^{2} T^{8} - 105 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 6 T + 99 T^{2} - 752 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 24 T + 365 T^{2} + 3368 T^{3} + 365 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 20567 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 55487 p^{2} T^{8} - 346 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 17 T + 205 T^{2} + 2138 T^{3} + 205 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 1179 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 95 T^{2} - 464 T^{3} + 95 p T^{4} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 98339 p^{2} T^{8} - 469 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.78947517201624457546203717413, −4.61426129428016841665820901654, −4.28603660174714292098384000101, −4.25514712357181279959073905360, −4.20538140682006614814992807258, −4.04186850934380039037985150141, −3.77113287934573649680881463495, −3.74079023681433928662615165964, −3.69632247988948049469051818247, −3.35293603993835951572298733275, −3.26144314545617586905436109413, −3.21504636662760560836523751796, −3.06578705105591175441893540715, −2.72345909511125749642439600407, −2.68340825366249047040082410771, −1.96195327317938833202945484473, −1.88794548920622952583164233758, −1.75507164389864830522404511013, −1.71003283218361743882007571138, −1.59840149186568241818746655758, −1.42130856727910920820615895387, −1.41824341864306792320228073102, −0.900374415020395066097176543167, −0.56404672460646161349956903855, −0.20083045312349155890648141678,
0.20083045312349155890648141678, 0.56404672460646161349956903855, 0.900374415020395066097176543167, 1.41824341864306792320228073102, 1.42130856727910920820615895387, 1.59840149186568241818746655758, 1.71003283218361743882007571138, 1.75507164389864830522404511013, 1.88794548920622952583164233758, 1.96195327317938833202945484473, 2.68340825366249047040082410771, 2.72345909511125749642439600407, 3.06578705105591175441893540715, 3.21504636662760560836523751796, 3.26144314545617586905436109413, 3.35293603993835951572298733275, 3.69632247988948049469051818247, 3.74079023681433928662615165964, 3.77113287934573649680881463495, 4.04186850934380039037985150141, 4.20538140682006614814992807258, 4.25514712357181279959073905360, 4.28603660174714292098384000101, 4.61426129428016841665820901654, 4.78947517201624457546203717413
Plot not available for L-functions of degree greater than 10.
