| L(s) = 1 | − 4·3-s − 16·5-s + 16·7-s + 2·9-s + 64·15-s − 64·21-s + 24·23-s + 140·25-s + 8·27-s − 256·35-s − 32·45-s + 132·49-s − 24·61-s + 32·63-s − 96·69-s − 560·75-s − 21·81-s + 32·83-s − 8·101-s + 1.02e3·105-s − 384·115-s − 10·121-s − 880·125-s + 127-s + 131-s − 128·135-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 7.15·5-s + 6.04·7-s + 2/3·9-s + 16.5·15-s − 13.9·21-s + 5.00·23-s + 28·25-s + 1.53·27-s − 43.2·35-s − 4.77·45-s + 18.8·49-s − 3.07·61-s + 4.03·63-s − 11.5·69-s − 64.6·75-s − 7/3·81-s + 3.51·83-s − 0.796·101-s + 99.9·105-s − 35.8·115-s − 0.909·121-s − 78.7·125-s + 0.0887·127-s + 0.0873·131-s − 11.0·135-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256679965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256679965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 7 | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 3 p T^{2} + 680 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 41 T^{2} + 960 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 24 T^{2} + 254 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 89 T^{2} + 3624 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 16 T^{2} + 1374 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 40 T^{2} + 2526 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 56 T^{2} + 3534 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 16 T^{2} - 1746 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 149 T^{2} + 9624 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 235 T^{2} + 25944 T^{4} - 235 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 89 | \( ( 1 - 248 T^{2} + 30606 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 25 T^{2} + 10368 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.79300659092178758734206315917, −3.78219246710198752725632890346, −3.58745638534452325420006951999, −3.42652389102651499912680496947, −3.32516166845337151180514295964, −3.25703917928437174134049074024, −3.11560974689396526928537178565, −3.11180808912276685604996208868, −2.92430343271531317903349941624, −2.85585450610263701283122100360, −2.56419545049002654765519262348, −2.27304465614043335533747562257, −2.27109615789880423473961935547, −2.19649423974671759756568413237, −1.90104758115410901003272714907, −1.49281398448168350416831181607, −1.48400024423664109149696023660, −1.37619800890587307028924820084, −1.33197582917163382497346347933, −0.941901876358425882173699966353, −0.821855632193230530375970708790, −0.78622312849425476741528679250, −0.52976261737729310981315160770, −0.38686004624533437059407399605, −0.27314947205492905358120193861,
0.27314947205492905358120193861, 0.38686004624533437059407399605, 0.52976261737729310981315160770, 0.78622312849425476741528679250, 0.821855632193230530375970708790, 0.941901876358425882173699966353, 1.33197582917163382497346347933, 1.37619800890587307028924820084, 1.48400024423664109149696023660, 1.49281398448168350416831181607, 1.90104758115410901003272714907, 2.19649423974671759756568413237, 2.27109615789880423473961935547, 2.27304465614043335533747562257, 2.56419545049002654765519262348, 2.85585450610263701283122100360, 2.92430343271531317903349941624, 3.11180808912276685604996208868, 3.11560974689396526928537178565, 3.25703917928437174134049074024, 3.32516166845337151180514295964, 3.42652389102651499912680496947, 3.58745638534452325420006951999, 3.78219246710198752725632890346, 3.79300659092178758734206315917
Plot not available for L-functions of degree greater than 10.
