| L(s) = 1 | + 486·3-s + 5.30e3·5-s − 3.88e4·7-s + 1.77e5·9-s − 1.04e6·11-s − 2.29e4·13-s + 2.57e6·15-s + 8.73e6·17-s + 7.34e6·19-s − 1.88e7·21-s + 6.71e6·23-s − 3.62e7·25-s + 5.73e7·27-s + 1.80e8·29-s + 2.11e8·31-s − 5.09e8·33-s − 2.06e8·35-s + 1.12e8·37-s − 1.11e7·39-s − 7.26e8·41-s + 2.16e8·43-s + 9.38e8·45-s − 2.17e9·47-s − 2.05e7·49-s + 4.24e9·51-s − 1.12e8·53-s − 5.55e9·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.758·5-s − 0.874·7-s + 9-s − 1.96·11-s − 0.0171·13-s + 0.875·15-s + 1.49·17-s + 0.680·19-s − 1.00·21-s + 0.217·23-s − 0.742·25-s + 0.769·27-s + 1.63·29-s + 1.32·31-s − 2.26·33-s − 0.663·35-s + 0.266·37-s − 0.0197·39-s − 0.979·41-s + 0.224·43-s + 0.758·45-s − 1.38·47-s − 0.0104·49-s + 1.72·51-s − 0.0367·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.661585424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.661585424\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 212 p^{2} T + 12869446 p T^{2} - 212 p^{13} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 38872 T + 1531611582 T^{2} + 38872 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 1047400 T + 582443349302 T^{2} + 1047400 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 22900 T + 3394284292494 T^{2} + 22900 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8733300 T + 76053146135686 T^{2} - 8733300 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7346600 T + 234273365886438 T^{2} - 7346600 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6711744 T + 1736184979908238 T^{2} - 6711744 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 180180692 T + 29060203717011374 T^{2} - 180180692 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 211581400 T + 49676382376667982 T^{2} - 211581400 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 112222700 T - 19046823489470754 T^{2} - 112222700 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 726961180 T + 1220821391254444982 T^{2} + 726961180 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 216408856 T + 301094542061252598 T^{2} - 216408856 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2174779088 T + 4972780192325352542 T^{2} + 2174779088 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 112024700 T + 2647443937277883614 T^{2} + 112024700 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3243949400 T + 1167855931192926038 T^{2} + 3243949400 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16526230620 T + \)\(15\!\cdots\!22\)\( T^{2} - 16526230620 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20772619112 T + \)\(26\!\cdots\!02\)\( T^{2} + 20772619112 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20637101600 T + \)\(45\!\cdots\!62\)\( T^{2} + 20637101600 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18548203500 T + \)\(70\!\cdots\!34\)\( T^{2} + 18548203500 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 28230083800 T + \)\(16\!\cdots\!78\)\( T^{2} - 28230083800 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7189282056 T - \)\(10\!\cdots\!82\)\( T^{2} - 7189282056 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 103679180788 T + \)\(82\!\cdots\!14\)\( T^{2} - 103679180788 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 199614486500 T + \)\(24\!\cdots\!26\)\( T^{2} - 199614486500 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−11.94186890229140191852579208187, −11.80270135747913305538350219443, −10.42336337751493201005217843481, −10.35706568527416399232434054083, −9.768114073061643199682863688701, −9.644382271379057591305484050169, −8.549931797276266671090817357418, −8.373073865348144355480063452193, −7.46661102506813603556292644287, −7.45018006591600342537753318669, −6.24663662265604381409837098299, −5.99824975682475272709985901126, −5.00491247171813884853596271196, −4.74984796918263732982703881495, −3.38026795817872120792712654190, −3.24849548781323506435633671558, −2.59534976518616244641478127638, −2.05714351411715187698840748518, −1.15946903199317709620295094021, −0.49209472351987691140298333060,
0.49209472351987691140298333060, 1.15946903199317709620295094021, 2.05714351411715187698840748518, 2.59534976518616244641478127638, 3.24849548781323506435633671558, 3.38026795817872120792712654190, 4.74984796918263732982703881495, 5.00491247171813884853596271196, 5.99824975682475272709985901126, 6.24663662265604381409837098299, 7.45018006591600342537753318669, 7.46661102506813603556292644287, 8.373073865348144355480063452193, 8.549931797276266671090817357418, 9.644382271379057591305484050169, 9.768114073061643199682863688701, 10.35706568527416399232434054083, 10.42336337751493201005217843481, 11.80270135747913305538350219443, 11.94186890229140191852579208187
