| L(s) = 1 | + 20·7-s + 16·11-s + 40·17-s − 40·23-s − 16·31-s + 200·41-s + 120·43-s + 200·49-s + 200·53-s − 312·61-s − 40·67-s + 80·71-s + 20·73-s + 320·77-s − 9·81-s − 240·83-s − 300·97-s + 232·101-s + 220·103-s − 160·107-s − 160·113-s + 800·119-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 20/7·7-s + 1.45·11-s + 2.35·17-s − 1.73·23-s − 0.516·31-s + 4.87·41-s + 2.79·43-s + 4.08·49-s + 3.77·53-s − 5.11·61-s − 0.597·67-s + 1.12·71-s + 0.273·73-s + 4.15·77-s − 1/9·81-s − 2.89·83-s − 3.09·97-s + 2.29·101-s + 2.13·103-s − 1.49·107-s − 1.41·113-s + 6.72·119-s − 0.198·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(16.80871023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.80871023\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1740 T^{3} + 13694 T^{4} - 1740 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 108 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 55678 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 19080 T^{3} + 419714 T^{4} - 19080 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 21126 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 1080 p T^{3} + 1442 p^{2} T^{4} + 1080 p^{3} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2096 T^{2} + 2222466 T^{4} - 2096 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 1338 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 3168578 T^{4} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 100 T + 5262 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 432120 T^{3} + 22864898 T^{4} - 432120 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2115554 T^{4} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1475400 T^{3} + 87973634 T^{4} - 1475400 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5816 T^{2} + 18089586 T^{4} - 5816 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 156 T + 12926 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 170280 T^{3} + 36190274 T^{4} + 170280 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 40 T + 882 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 46140 T^{3} + 1512014 T^{4} - 46140 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22612 T^{2} + 204343398 T^{4} - 22612 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 240 T + 28800 T^{2} + 2549040 T^{3} + 211683458 T^{4} + 2549040 p^{2} T^{5} + 28800 p^{4} T^{6} + 240 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 11716 T^{2} + 151160646 T^{4} + 11716 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 300 T + 45000 T^{2} + 5967300 T^{3} + 681431438 T^{4} + 5967300 p^{2} T^{5} + 45000 p^{4} T^{6} + 300 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.06799467242259103279023226597, −6.28645157402921710735504303515, −6.24183557027119255213741613547, −5.98984847145059838005657110661, −5.87485542425497325434031468932, −5.62876967100398887823409960853, −5.50941904657791516252109867042, −5.34625930666839855881951610136, −4.92894297191316099984527433320, −4.57066846018239856135781362346, −4.36460380462326865425964409903, −4.31652864056543687701961546236, −4.00868297928116944703372235013, −3.95127075558606183119643194232, −3.72249922097480972227700420952, −3.01420071208612756990260548904, −3.01052618829453569147523090559, −2.50340015227475133104046525077, −2.38784960880184873483644749360, −2.01609492896569505111277336682, −1.53370119436264792630276150082, −1.37034183435050112049221645310, −1.27346393831714704793864979376, −0.66238431173483204100224253549, −0.64970099689296551964658317350,
0.64970099689296551964658317350, 0.66238431173483204100224253549, 1.27346393831714704793864979376, 1.37034183435050112049221645310, 1.53370119436264792630276150082, 2.01609492896569505111277336682, 2.38784960880184873483644749360, 2.50340015227475133104046525077, 3.01052618829453569147523090559, 3.01420071208612756990260548904, 3.72249922097480972227700420952, 3.95127075558606183119643194232, 4.00868297928116944703372235013, 4.31652864056543687701961546236, 4.36460380462326865425964409903, 4.57066846018239856135781362346, 4.92894297191316099984527433320, 5.34625930666839855881951610136, 5.50941904657791516252109867042, 5.62876967100398887823409960853, 5.87485542425497325434031468932, 5.98984847145059838005657110661, 6.24183557027119255213741613547, 6.28645157402921710735504303515, 7.06799467242259103279023226597
