| L(s) = 1 | + 32·2-s + 256·4-s − 8.19e3·8-s + 3.66e4·9-s − 1.87e4·11-s − 2.62e5·16-s + 1.17e6·18-s − 6.00e5·22-s + 1.95e6·23-s + 5.35e5·25-s − 1.72e7·29-s − 2.09e6·32-s + 9.37e6·36-s − 5.42e6·37-s − 1.39e8·43-s − 4.80e6·44-s + 6.26e7·46-s + 1.71e7·50-s − 1.36e8·53-s − 5.52e8·58-s + 5.03e7·64-s − 4.85e8·67-s − 3.77e8·71-s − 2.99e8·72-s − 1.73e8·74-s − 1.35e9·79-s + 3.87e8·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s + 1.86·9-s − 0.386·11-s − 16-s + 2.63·18-s − 0.546·22-s + 1.45·23-s + 0.273·25-s − 4.53·29-s − 0.353·32-s + 0.930·36-s − 0.476·37-s − 6.20·43-s − 0.193·44-s + 2.06·46-s + 0.387·50-s − 2.36·53-s − 6.41·58-s + 3/8·64-s − 2.94·67-s − 1.76·71-s − 1.31·72-s − 0.673·74-s − 3.91·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.798807255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798807255\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 36614 T^{2} + 953164507 T^{4} - 36614 p^{18} T^{6} + p^{36} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 21402 p^{2} T^{2} - 5645470021 p^{4} T^{4} - 21402 p^{20} T^{6} + p^{36} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 9380 T - 2269963291 T^{2} + 9380 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 10847863142 T^{2} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 227593410082 T^{2} + \)\(37\!\cdots\!15\)\( T^{4} - 227593410082 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 328706803750 T^{2} + \)\(39\!\cdots\!59\)\( T^{4} - 328706803750 p^{18} T^{6} + p^{36} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 978936 T - 842836969367 T^{2} - 978936 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4317214 T + p^{9} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 10714676730050 T^{2} - \)\(58\!\cdots\!41\)\( T^{4} + 10714676730050 p^{18} T^{6} + p^{36} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 73362 p T - 89549894089 p^{2} T^{2} + 73362 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 573721268650450 T^{2} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 34755692 T + p^{9} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 459315263762526 T^{2} - \)\(10\!\cdots\!13\)\( T^{4} - 459315263762526 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 68067926 T + 1333478958139343 T^{2} + 68067926 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 16154177553222390 T^{2} + \)\(18\!\cdots\!79\)\( T^{4} - 16154177553222390 p^{18} T^{6} + p^{36} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 4112136386785286 T^{2} - \)\(11\!\cdots\!85\)\( T^{4} + 4112136386785286 p^{18} T^{6} + p^{36} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 242944420 T + 31815456812841453 T^{2} + 242944420 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 94292464 T + p^{9} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 99965099610294034 T^{2} + \)\(65\!\cdots\!87\)\( T^{4} - 99965099610294034 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 677625160 T + 339324261482407281 T^{2} + 677625160 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 179122938928854694 T^{2} + p^{18} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 392379595356141618 T^{2} + \)\(31\!\cdots\!43\)\( T^{4} - 392379595356141618 p^{18} T^{6} + p^{36} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 89760987702549566 T^{2} + p^{18} T^{4} )^{2} \) |
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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
−8.320863445627822563897459330694, −8.181149747729665248080621555502, −7.68508868613057131397510518469, −7.24430605290234545539935876893, −7.11195215711600678956810975526, −6.91358399546534280101534851732, −6.81803619496454863687225560664, −5.95964825128259852391478032835, −5.95332898082555041584129335118, −5.59340208504674098739267949044, −5.28956668993597371483952606418, −4.75723255860385345961392968191, −4.63666548033610781262074518017, −4.59523504397545106780705477622, −3.99753945937354567011336196058, −3.67958321656040075682767319744, −3.27688060155362905524504852099, −3.05127886306947498739677852048, −3.01341904353913661558346519914, −1.95845035759730515852111245670, −1.63224912591420990443847144413, −1.62408255641413359148298240302, −1.42857469775110780677427407856, −0.38307665517392088923735047424, −0.18459320591626713600076540985,
0.18459320591626713600076540985, 0.38307665517392088923735047424, 1.42857469775110780677427407856, 1.62408255641413359148298240302, 1.63224912591420990443847144413, 1.95845035759730515852111245670, 3.01341904353913661558346519914, 3.05127886306947498739677852048, 3.27688060155362905524504852099, 3.67958321656040075682767319744, 3.99753945937354567011336196058, 4.59523504397545106780705477622, 4.63666548033610781262074518017, 4.75723255860385345961392968191, 5.28956668993597371483952606418, 5.59340208504674098739267949044, 5.95332898082555041584129335118, 5.95964825128259852391478032835, 6.81803619496454863687225560664, 6.91358399546534280101534851732, 7.11195215711600678956810975526, 7.24430605290234545539935876893, 7.68508868613057131397510518469, 8.181149747729665248080621555502, 8.320863445627822563897459330694
