| L(s) = 1 | − 5.88e3·3-s − 6.04e5·5-s − 2.53e7·7-s + 1.12e8·9-s + 1.25e9·11-s + 1.32e9·13-s + 3.55e9·15-s − 2.74e10·17-s + 1.01e11·19-s + 1.49e11·21-s − 1.34e11·23-s − 8.04e11·25-s − 1.88e12·27-s − 2.33e12·29-s − 2.78e11·31-s − 7.40e12·33-s + 1.53e13·35-s − 2.09e13·37-s − 7.76e12·39-s + 4.16e12·41-s − 1.11e14·43-s − 6.80e13·45-s − 1.96e14·47-s + 1.38e14·49-s + 1.61e14·51-s + 4.87e14·53-s − 7.60e14·55-s + ⋯ |
| L(s) = 1 | − 0.517·3-s − 0.691·5-s − 1.66·7-s + 0.872·9-s + 1.77·11-s + 0.448·13-s + 0.357·15-s − 0.956·17-s + 1.36·19-s + 0.859·21-s − 0.358·23-s − 1.05·25-s − 1.28·27-s − 0.867·29-s − 0.0587·31-s − 0.916·33-s + 1.14·35-s − 0.979·37-s − 0.232·39-s + 0.0814·41-s − 1.45·43-s − 0.603·45-s − 1.20·47-s + 0.595·49-s + 0.494·51-s + 1.07·53-s − 1.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.1231343986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1231343986\) |
| \(L(\frac{19}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 1960 p T - 321646 p^{5} T^{2} + 1960 p^{18} T^{3} + p^{34} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 604044 T + 46794698206 p^{2} T^{2} + 604044 p^{17} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 25350160 T + 10288992992430 p^{2} T^{2} + 25350160 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 114513480 p T + 7489675618956502 p^{2} T^{2} - 114513480 p^{18} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1320052580 T + 661181623540177494 p T^{2} - 1320052580 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 27498226140 T + \)\(18\!\cdots\!58\)\( T^{2} + 27498226140 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 101133633832 T + \)\(10\!\cdots\!34\)\( T^{2} - 101133633832 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 134767491120 T + \)\(21\!\cdots\!70\)\( T^{2} + 134767491120 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2337155582652 T + \)\(94\!\cdots\!94\)\( T^{2} + 2337155582652 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 278836113472 T + \)\(39\!\cdots\!18\)\( T^{2} + 278836113472 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 20929802888140 T + \)\(93\!\cdots\!10\)\( T^{2} + 20929802888140 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4166592315732 T + \)\(39\!\cdots\!18\)\( T^{2} - 4166592315732 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 111143148534440 T + \)\(13\!\cdots\!86\)\( T^{2} + 111143148534440 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 196772651157600 T + \)\(53\!\cdots\!18\)\( T^{2} + 196772651157600 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 487965122736660 T + \)\(41\!\cdots\!50\)\( T^{2} - 487965122736660 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2835904197813624 T + \)\(45\!\cdots\!82\)\( T^{2} - 2835904197813624 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 67544034994436 T - \)\(59\!\cdots\!34\)\( T^{2} - 67544034994436 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 995171321546360 T + \)\(18\!\cdots\!70\)\( T^{2} + 995171321546360 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3882245493215376 T + \)\(21\!\cdots\!26\)\( T^{2} + 3882245493215376 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12746425881769580 T + \)\(13\!\cdots\!82\)\( T^{2} + 12746425881769580 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14984271534065504 T + \)\(37\!\cdots\!22\)\( T^{2} - 14984271534065504 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 43899417809893800 T + \)\(13\!\cdots\!30\)\( T^{2} - 43899417809893800 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 51909007958846388 T + \)\(34\!\cdots\!94\)\( T^{2} - 51909007958846388 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 49281007789848380 T + \)\(91\!\cdots\!10\)\( T^{2} + 49281007789848380 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−11.75209488895277369347877962574, −11.57469904515428493448299097562, −10.70125386851666732483686004112, −9.935195798990259663776933315242, −9.641570479825750773834077817300, −9.198976922505854294244444384080, −8.523778050391109656655437118918, −7.70414015126773071290713269434, −6.93515471218173662220950292363, −6.83818217679408536918818334927, −6.14573249430812669948222060831, −5.64165836863978080671605980794, −4.76256331627358615967237912471, −3.93903574521571284714471194645, −3.64484472829096702483903811553, −3.36743300850192477173239880511, −2.15470922033901820123202950111, −1.54838951235198962646549262139, −0.916955802743919316728846456252, −0.091255443207438614412718979520,
0.091255443207438614412718979520, 0.916955802743919316728846456252, 1.54838951235198962646549262139, 2.15470922033901820123202950111, 3.36743300850192477173239880511, 3.64484472829096702483903811553, 3.93903574521571284714471194645, 4.76256331627358615967237912471, 5.64165836863978080671605980794, 6.14573249430812669948222060831, 6.83818217679408536918818334927, 6.93515471218173662220950292363, 7.70414015126773071290713269434, 8.523778050391109656655437118918, 9.198976922505854294244444384080, 9.641570479825750773834077817300, 9.935195798990259663776933315242, 10.70125386851666732483686004112, 11.57469904515428493448299097562, 11.75209488895277369347877962574
