| L(s) = 1 | + 70·3-s + 1.55e3·5-s + 4.80e3·7-s − 2.44e4·9-s − 6.23e4·11-s + 1.22e5·13-s + 1.08e5·15-s + 7.35e4·17-s − 1.17e6·19-s + 3.36e5·21-s − 2.26e6·23-s + 1.95e6·25-s − 2.39e6·27-s − 1.92e6·29-s − 2.97e6·31-s − 4.36e6·33-s + 7.46e6·35-s − 1.34e7·37-s + 8.59e6·39-s − 3.63e7·41-s + 2.19e7·43-s − 3.80e7·45-s + 1.36e6·47-s + 1.72e7·49-s + 5.15e6·51-s − 1.78e7·53-s − 9.69e7·55-s + ⋯ |
| L(s) = 1 | + 0.498·3-s + 1.11·5-s + 0.755·7-s − 1.24·9-s − 1.28·11-s + 1.19·13-s + 0.554·15-s + 0.213·17-s − 2.06·19-s + 0.377·21-s − 1.68·23-s + 0.999·25-s − 0.866·27-s − 0.504·29-s − 0.579·31-s − 0.641·33-s + 0.840·35-s − 1.17·37-s + 0.594·39-s − 2.00·41-s + 0.979·43-s − 1.38·45-s + 0.0407·47-s + 3/7·49-s + 0.106·51-s − 0.311·53-s − 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 70 T + 9794 p T^{2} - 70 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 1554 T + 92706 p T^{2} - 1554 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 62388 T + 5317674102 T^{2} + 62388 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 122766 T + 1914771578 p T^{2} - 122766 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 73584 T + 138587763294 T^{2} - 73584 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 61642 p T + 942609653910 T^{2} + 61642 p^{10} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2262384 T + 185222048898 p T^{2} + 2262384 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1923360 T + 23001703439382 T^{2} + 1923360 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2977884 T + 31127734980830 T^{2} + 2977884 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13418528 T + 97243912721094 T^{2} + 13418528 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 36367800 T + 982922223862366 T^{2} + 36367800 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 510812 p T + 1094756328321366 T^{2} - 510812 p^{10} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1362732 T + 1922701259198590 T^{2} - 1362732 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 17898612 T + 1151999595207502 T^{2} + 17898612 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 224710542 T + 28856421320644278 T^{2} + 224710542 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 85847118 T + 5849816165467562 T^{2} + 85847118 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 179568872 T + 37296819148445334 T^{2} + 179568872 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 231378168 T + 105070505575629582 T^{2} + 231378168 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 88098332 T + 119680894964975046 T^{2} - 88098332 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 184274184 T + 157166499568643102 T^{2} - 184274184 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 624641094 T + 301231392696541014 T^{2} + 624641094 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1574777148 T + 1271578866761386870 T^{2} + 1574777148 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 213665984 T + 1447970079084154398 T^{2} - 213665984 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.37211960482704135705723988769, −10.99193285055328845762613989646, −10.44281673173423532926156087770, −10.24754384706212476328066882176, −9.267797019475163818678256625811, −8.820419105710141973854788349581, −8.227897422334648374966341037810, −8.195351295409218943987048474653, −7.26292710836075734876161836585, −6.35836500095478997134958770491, −5.84566678903723056886336692934, −5.59117322943696552248326465461, −4.76342857029709052640502079722, −4.02867133881979766326343764035, −3.15244372213429581802583135762, −2.60089364871424124587013638431, −1.72745791280549328105950550392, −1.70219529909319786190418957379, 0, 0,
1.70219529909319786190418957379, 1.72745791280549328105950550392, 2.60089364871424124587013638431, 3.15244372213429581802583135762, 4.02867133881979766326343764035, 4.76342857029709052640502079722, 5.59117322943696552248326465461, 5.84566678903723056886336692934, 6.35836500095478997134958770491, 7.26292710836075734876161836585, 8.195351295409218943987048474653, 8.227897422334648374966341037810, 8.820419105710141973854788349581, 9.267797019475163818678256625811, 10.24754384706212476328066882176, 10.44281673173423532926156087770, 10.99193285055328845762613989646, 11.37211960482704135705723988769
