| L(s) = 1 | − 14·3-s − 42·5-s − 146·9-s + 716·11-s + 714·13-s + 588·15-s + 1.34e3·17-s − 1.94e3·19-s + 1.79e3·23-s − 102·25-s + 3.43e3·27-s − 1.20e3·29-s − 6.80e3·31-s − 1.00e4·33-s + 1.46e4·37-s − 9.99e3·39-s − 7.89e3·41-s − 524·43-s + 6.13e3·45-s − 1.83e4·47-s − 1.88e4·51-s + 4.51e4·53-s − 3.00e4·55-s + 2.72e4·57-s + 2.25e4·59-s + 5.28e4·61-s − 2.99e4·65-s + ⋯ |
| L(s) = 1 | − 0.898·3-s − 0.751·5-s − 0.600·9-s + 1.78·11-s + 1.17·13-s + 0.674·15-s + 1.12·17-s − 1.23·19-s + 0.706·23-s − 0.0326·25-s + 0.905·27-s − 0.264·29-s − 1.27·31-s − 1.60·33-s + 1.75·37-s − 1.05·39-s − 0.733·41-s − 0.0432·43-s + 0.451·45-s − 1.21·47-s − 1.01·51-s + 2.20·53-s − 1.34·55-s + 1.11·57-s + 0.844·59-s + 1.81·61-s − 0.880·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.380266718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.380266718\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 14 T + 38 p^{2} T^{2} + 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 42 T + 1866 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 716 T + 412438 T^{2} - 716 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 714 T + 814258 T^{2} - 714 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1344 T + 2728510 T^{2} - 1344 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1946 T + 4229670 T^{2} + 1946 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1792 T + 11254510 T^{2} - 1792 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1200 T - 16154090 T^{2} + 1200 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6804 T + 30441118 T^{2} + 6804 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14640 T + 187693126 T^{2} - 14640 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7896 T + 209593854 T^{2} + 7896 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 524 T + 242298998 T^{2} + 524 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18396 T + 435885630 T^{2} + 18396 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 45132 T + 1149514990 T^{2} - 45132 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 22582 T + 1531622854 T^{2} - 22582 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 52822 T + 2145929546 T^{2} - 52822 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9848 T + 2714812022 T^{2} + 9848 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 840 T + 427300302 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 122052 T + 7590539974 T^{2} - 122052 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 31704 T + 6042098590 T^{2} + 31704 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 36974 T + 4839605030 T^{2} - 36974 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 210588 T + 21813719542 T^{2} - 210588 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 44240 T + 2438219582 T^{2} - 44240 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.578982954577354140998459930130, −9.451829956363657024653924548405, −8.693338469435825687156315515797, −8.617672310747782743644422657205, −8.099598967670141187943732271770, −7.65982384080178357680129619593, −6.92575044714805092815355437719, −6.72950774177481179857760282477, −6.14474069399899338652475443250, −5.99238098266184798073509570135, −5.18186134089938980220879483256, −5.14689426228218125915424948791, −4.03419559650972499618254299015, −3.89987593956695652274807072993, −3.61453825582208525683394019018, −2.83221284806293859105693043651, −2.07029295629872772380746580973, −1.39365303183960622037605298210, −0.77119680208208657256980920474, −0.48807134220357997457458023684,
0.48807134220357997457458023684, 0.77119680208208657256980920474, 1.39365303183960622037605298210, 2.07029295629872772380746580973, 2.83221284806293859105693043651, 3.61453825582208525683394019018, 3.89987593956695652274807072993, 4.03419559650972499618254299015, 5.14689426228218125915424948791, 5.18186134089938980220879483256, 5.99238098266184798073509570135, 6.14474069399899338652475443250, 6.72950774177481179857760282477, 6.92575044714805092815355437719, 7.65982384080178357680129619593, 8.099598967670141187943732271770, 8.617672310747782743644422657205, 8.693338469435825687156315515797, 9.451829956363657024653924548405, 9.578982954577354140998459930130
