| L(s) = 1 | + 660·5-s + 3.48e4·9-s − 3.48e4·11-s − 2.27e5·19-s + 1.16e5·25-s + 6.26e6·29-s + 3.74e5·31-s − 1.76e7·41-s + 2.29e7·45-s + 1.53e8·49-s − 2.29e7·55-s + 4.38e8·59-s − 1.03e8·61-s − 5.04e8·71-s + 1.79e9·79-s + 3.16e8·81-s + 2.38e9·89-s − 1.50e8·95-s − 1.21e9·99-s − 1.46e9·101-s + 2.38e9·109-s − 3.67e9·121-s + 1.15e9·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.472·5-s + 1.77·9-s − 0.716·11-s − 0.400·19-s + 0.0598·25-s + 1.64·29-s + 0.0728·31-s − 0.975·41-s + 0.836·45-s + 3.79·49-s − 0.338·55-s + 4.71·59-s − 0.952·61-s − 2.35·71-s + 5.18·79-s + 0.817·81-s + 4.02·89-s − 0.189·95-s − 1.26·99-s − 1.39·101-s + 1.61·109-s − 1.55·121-s + 0.423·125-s + 0.776·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.026979668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.026979668\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 5 | $D_{4}$ | \( 1 - 132 p T + 102 p^{5} T^{2} - 132 p^{10} T^{3} + p^{18} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 34844 T^{2} + 99693718 p^{2} T^{4} - 34844 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 153156620 T^{2} + 186133228268502 p^{2} T^{4} - 153156620 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 17400 T + 2292818982 T^{2} + 17400 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 14000436724 T^{2} + 83025702170976147702 T^{4} - 14000436724 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 181978394436 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} - 181978394436 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 113832 T + 402567657014 T^{2} + 113832 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 820282526580 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} + 820282526580 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 3132828 T + 12854589500734 T^{2} - 3132828 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 187232 T + 40099942836798 T^{2} - 187232 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 462603258659540 T^{2} + \)\(87\!\cdots\!58\)\( T^{4} - 462603258659540 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8824068 T + 671552976228678 T^{2} + 8824068 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 358707990293372 T^{2} + \)\(21\!\cdots\!94\)\( T^{4} - 358707990293372 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 1037666802860076 T^{2} + \)\(16\!\cdots\!22\)\( T^{4} - 1037666802860076 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 5310844341928340 T^{2} + \)\(28\!\cdots\!78\)\( T^{4} - 5310844341928340 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 219497736 T + 28852098295168902 T^{2} - 219497736 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 51522236 T - 2665246277981394 T^{2} + 51522236 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 83417417389239260 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - 83417417389239260 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 252040944 T + 101042798798695246 T^{2} + 252040944 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 79546297979572004 T^{2} + \)\(52\!\cdots\!42\)\( T^{4} - 79546297979572004 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 897477504 T + 421888354983619742 T^{2} - 897477504 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 186261173343564060 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - 186261173343564060 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1190659092 T + 825013495390166934 T^{2} - 1190659092 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 899946211039106180 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} - 899946211039106180 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.98351115407716587155092486028, −11.24849808607303807519580168606, −10.67511409237740328380888084798, −10.40696779093953260559744388326, −10.18418884109282435000271511635, −10.09462104451817059545417671303, −9.393034218773736053742133713174, −9.063392337495464911347893186160, −8.579167606187041829929906428618, −8.306664121623483277076990165800, −7.53572376205453944662711563450, −7.45762597644838548312978382175, −6.93584124157412999645812275227, −6.44260802503671268791772635036, −6.18237397002615047002713785311, −5.31349708618594896827777726877, −5.12610363506469404371186895216, −4.57230197280096095532331445340, −3.90062114571439388228656887930, −3.71092890107768823603798656780, −2.62898624204244308987092990620, −2.30817244625680916472162502770, −1.71863852892113351256218250222, −0.78172087094473610666326816353, −0.74568161285112177365677040010,
0.74568161285112177365677040010, 0.78172087094473610666326816353, 1.71863852892113351256218250222, 2.30817244625680916472162502770, 2.62898624204244308987092990620, 3.71092890107768823603798656780, 3.90062114571439388228656887930, 4.57230197280096095532331445340, 5.12610363506469404371186895216, 5.31349708618594896827777726877, 6.18237397002615047002713785311, 6.44260802503671268791772635036, 6.93584124157412999645812275227, 7.45762597644838548312978382175, 7.53572376205453944662711563450, 8.306664121623483277076990165800, 8.579167606187041829929906428618, 9.063392337495464911347893186160, 9.393034218773736053742133713174, 10.09462104451817059545417671303, 10.18418884109282435000271511635, 10.40696779093953260559744388326, 10.67511409237740328380888084798, 11.24849808607303807519580168606, 11.98351115407716587155092486028
