| L(s) = 1 | + 6·3-s + 82·5-s + 98·7-s − 114·9-s − 340·11-s + 910·13-s + 492·15-s + 3.21e3·17-s + 674·19-s + 588·21-s + 1.10e3·23-s + 1.89e3·25-s − 126·27-s + 8.06e3·29-s + 6.21e3·31-s − 2.04e3·33-s + 8.03e3·35-s − 8.51e3·37-s + 5.46e3·39-s − 1.30e3·41-s + 1.00e4·43-s − 9.34e3·45-s + 1.27e4·47-s + 7.20e3·49-s + 1.92e4·51-s − 1.12e4·53-s − 2.78e4·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.46·5-s + 0.755·7-s − 0.469·9-s − 0.847·11-s + 1.49·13-s + 0.564·15-s + 2.69·17-s + 0.428·19-s + 0.290·21-s + 0.435·23-s + 0.607·25-s − 0.0332·27-s + 1.78·29-s + 1.16·31-s − 0.326·33-s + 1.10·35-s − 1.02·37-s + 0.574·39-s − 0.121·41-s + 0.825·43-s − 0.688·45-s + 0.841·47-s + 3/7·49-s + 1.03·51-s − 0.548·53-s − 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.724156304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.724156304\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 p T + 50 p T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 82 T + 4826 T^{2} - 82 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 340 T + 338582 T^{2} + 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 70 p T + 2514 p^{2} T^{2} - 70 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3216 T + 5412958 T^{2} - 3216 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 674 T + 4367142 T^{2} - 674 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 48 p T + 495170 p T^{2} - 48 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8064 T + 52795702 T^{2} - 8064 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6212 T + 51691038 T^{2} - 6212 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8512 T + 104326950 T^{2} + 8512 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1304 T + 73546526 T^{2} + 1304 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10004 T + 99339510 T^{2} - 10004 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12748 T + 323438270 T^{2} - 12748 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11220 T + 664373806 T^{2} + 11220 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12018 T - 285266426 T^{2} - 12018 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 102738 T + 4326026138 T^{2} - 102738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24136 T + 1542084918 T^{2} + 24136 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 89720 T + 4576356302 T^{2} + 89720 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 55588 T + 3902743302 T^{2} + 55588 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 48824 T + 5430110622 T^{2} + 48824 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 35782 T + 4098945062 T^{2} + 35782 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18300 T + 3539716918 T^{2} + 18300 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 69984 T + 18325482398 T^{2} + 69984 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.00265354754268158607019121227, −12.57534483926081382056934236605, −11.69349054183787082977052529108, −11.61937122265126939252123601461, −10.49773667111504950591165934186, −10.30854055502639577000636834451, −9.956527772253181085534124449769, −9.175644737350158140734299292197, −8.389059609708276845641752414668, −8.352709442719616720713951114048, −7.56753035858798035479234531490, −6.78852885878997422773713587067, −5.89544904508370381566828272981, −5.59478774042794742190795859122, −5.15570802566193394091955554311, −4.05904731979819345079837105692, −3.03014646403220171801072221759, −2.64647660339951618211486692437, −1.38719767701399037352558598127, −1.05333080012065916011909484369,
1.05333080012065916011909484369, 1.38719767701399037352558598127, 2.64647660339951618211486692437, 3.03014646403220171801072221759, 4.05904731979819345079837105692, 5.15570802566193394091955554311, 5.59478774042794742190795859122, 5.89544904508370381566828272981, 6.78852885878997422773713587067, 7.56753035858798035479234531490, 8.352709442719616720713951114048, 8.389059609708276845641752414668, 9.175644737350158140734299292197, 9.956527772253181085534124449769, 10.30854055502639577000636834451, 10.49773667111504950591165934186, 11.61937122265126939252123601461, 11.69349054183787082977052529108, 12.57534483926081382056934236605, 13.00265354754268158607019121227
