| L(s) = 1 | + 108·9-s − 96·29-s − 6.96e3·41-s − 1.66e4·49-s − 1.80e3·61-s + 7.29e3·81-s − 5.89e4·89-s + 3.63e4·101-s − 3.44e3·109-s + 8.80e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.18e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 0.114·29-s − 4.14·41-s − 6.92·49-s − 0.485·61-s + 10/9·81-s − 7.44·89-s + 3.56·101-s − 0.289·109-s + 6.01·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 4.14·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(18.10985000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.10985000\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 8308 T^{2} + 28644198 T^{4} + 8308 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 44020 T^{2} + 862213542 T^{4} - 44020 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 13 | \( ( 1 - 59132 T^{2} + 2002404678 T^{4} - 59132 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 - 241852 T^{2} + 26454460038 T^{4} - 241852 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 - 202468 T^{2} + 27737056518 T^{4} - 202468 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 23 | \( ( 1 + 975940 T^{2} + 394084152582 T^{4} + 975940 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 + 24 T + 1362686 T^{2} + 24 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 31 | \( ( 1 - 2595364 T^{2} + 3349044360006 T^{4} - 2595364 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 5944604 T^{2} + 15854519975046 T^{4} - 5944604 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 + 1740 T + 3816422 T^{2} + 1740 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4112740 T^{2} + 14250310310022 T^{4} + 4112740 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4187132 T^{2} + 14113409985798 T^{4} - 4187132 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 - 25843036 T^{2} + 289396670757126 T^{4} - 25843036 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 40267060 T^{2} + 686890438115622 T^{4} - 40267060 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 + 452 T + 13428438 T^{2} + 452 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 67 | \( ( 1 + 69140068 T^{2} + 1976862971872518 T^{4} + 69140068 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 43327300 T^{2} + 1348103415822342 T^{4} - 43327300 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 - 88789532 T^{2} + 3581552025133638 T^{4} - 88789532 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 - 121704868 T^{2} + 6477469249695558 T^{4} - 121704868 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 83 | \( ( 1 + 116513380 T^{2} + 7645245945348102 T^{4} + 116513380 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 + 14748 T + 178529078 T^{2} + 14748 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 97 | \( ( 1 - 157799996 T^{2} + 12311450620225926 T^{4} - 157799996 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.47053331179420281071777039539, −3.36666492990857647795509628425, −3.28448989437252107975244915468, −3.27447007588382609127747500829, −3.12340184121475477509118025052, −2.88911933148912000195665123925, −2.77150001224583543157828145971, −2.72111967021937291963939615559, −2.66377629209723775916151706278, −2.19718828056832393095385973843, −2.06682594099858431960061796028, −2.01477724442523218088305188520, −1.99682363431815040736078255626, −1.57204986391629156694789715137, −1.57012123072869500638462208965, −1.53738228844290186962839873264, −1.45960193980397303002508190433, −1.38667858300587673079738887027, −1.28527510373151655717381964958, −0.75400165807047587690607922786, −0.67866149906019942055895336764, −0.39770580047786612164973405896, −0.37329939222889162107016088554, −0.33099165812088184885120601904, −0.27880893221710097782413646458,
0.27880893221710097782413646458, 0.33099165812088184885120601904, 0.37329939222889162107016088554, 0.39770580047786612164973405896, 0.67866149906019942055895336764, 0.75400165807047587690607922786, 1.28527510373151655717381964958, 1.38667858300587673079738887027, 1.45960193980397303002508190433, 1.53738228844290186962839873264, 1.57012123072869500638462208965, 1.57204986391629156694789715137, 1.99682363431815040736078255626, 2.01477724442523218088305188520, 2.06682594099858431960061796028, 2.19718828056832393095385973843, 2.66377629209723775916151706278, 2.72111967021937291963939615559, 2.77150001224583543157828145971, 2.88911933148912000195665123925, 3.12340184121475477509118025052, 3.27447007588382609127747500829, 3.28448989437252107975244915468, 3.36666492990857647795509628425, 3.47053331179420281071777039539
Plot not available for L-functions of degree greater than 10.
