| L(s) = 1 | − 5·2-s + 8·4-s + 5·8-s − 68·11-s − 25·16-s + 340·22-s − 40·23-s + 125·25-s + 332·29-s + 40·32-s − 450·37-s − 360·43-s − 544·44-s + 200·46-s − 625·50-s + 590·53-s − 1.66e3·58-s − 487·64-s + 740·67-s − 1.37e3·71-s + 2.25e3·74-s + 1.38e3·79-s + 1.80e3·86-s − 340·88-s − 320·92-s + 1.00e3·100-s − 2.95e3·106-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 4-s + 0.220·8-s − 1.86·11-s − 0.390·16-s + 3.29·22-s − 0.362·23-s + 25-s + 2.12·29-s + 0.220·32-s − 1.99·37-s − 1.27·43-s − 1.86·44-s + 0.641·46-s − 1.76·50-s + 1.52·53-s − 3.75·58-s − 0.951·64-s + 1.34·67-s − 2.30·71-s + 3.53·74-s + 1.97·79-s + 2.25·86-s − 0.411·88-s − 0.362·92-s + 100-s − 2.70·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3429144782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3429144782\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T + 17 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 68 T + 3293 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T - 10567 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 450 T + 151847 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 590 T + 199223 T^{2} - 590 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 740 T + 246837 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 688 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1384 T + 1422417 T^{2} - 1384 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−10.72488346238157930997773090150, −10.27649055342522064769704212857, −10.05913350975079701122932665626, −9.710361886567874244111428662946, −8.889780367959263699654933175021, −8.691118422527247735811299287186, −8.390599288122067433831491335047, −7.88750820645534726855723970240, −7.56267671468386055070745546481, −6.85999101899748916062975684564, −6.57075982543942919019653135865, −5.72842454974206573142089391840, −5.06251122322544947950230412996, −4.93235381803303617779544136860, −4.04039582634859740243095377758, −3.13981837437279177474721732602, −2.69183637303999692990359341430, −1.87588901411534692339525092058, −0.991250718959446163854458220873, −0.31981083005606228011148247347,
0.31981083005606228011148247347, 0.991250718959446163854458220873, 1.87588901411534692339525092058, 2.69183637303999692990359341430, 3.13981837437279177474721732602, 4.04039582634859740243095377758, 4.93235381803303617779544136860, 5.06251122322544947950230412996, 5.72842454974206573142089391840, 6.57075982543942919019653135865, 6.85999101899748916062975684564, 7.56267671468386055070745546481, 7.88750820645534726855723970240, 8.390599288122067433831491335047, 8.691118422527247735811299287186, 8.889780367959263699654933175021, 9.710361886567874244111428662946, 10.05913350975079701122932665626, 10.27649055342522064769704212857, 10.72488346238157930997773090150
