| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5264837164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5264837164\) |
| \(L(\frac{11}{2})\) |
|
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
−8.813425062424795752683403288510, −8.599904983945699453584266043853, −8.085122379401239737275337086690, −7.50061760722133336860926575574, −7.49550758068535176037181982847, −7.44925800331540740854881078062, −6.70072687890016214101399289302, −6.58796414799344235750517306120, −6.43319577994854381794231388830, −5.78704895454427203815381067693, −5.73373542922201765317671432967, −5.21764260082796778816890480116, −4.62214577850567925568235791303, −4.55703950750709432623069486376, −4.37260265610026854118288041123, −3.73859272757801641443997109666, −3.58617837980183997232772471811, −2.89777878375882513676826471407, −2.81881620349658783116104644295, −2.06921616857292878043779532604, −1.92785621481345747023979783507, −1.25679013466060857457306409811, −1.13895523049737474680580105878, −0.974382014940247628811505185061, −0.07820571377745368488940178091,
0.07820571377745368488940178091, 0.974382014940247628811505185061, 1.13895523049737474680580105878, 1.25679013466060857457306409811, 1.92785621481345747023979783507, 2.06921616857292878043779532604, 2.81881620349658783116104644295, 2.89777878375882513676826471407, 3.58617837980183997232772471811, 3.73859272757801641443997109666, 4.37260265610026854118288041123, 4.55703950750709432623069486376, 4.62214577850567925568235791303, 5.21764260082796778816890480116, 5.73373542922201765317671432967, 5.78704895454427203815381067693, 6.43319577994854381794231388830, 6.58796414799344235750517306120, 6.70072687890016214101399289302, 7.44925800331540740854881078062, 7.49550758068535176037181982847, 7.50061760722133336860926575574, 8.085122379401239737275337086690, 8.599904983945699453584266043853, 8.813425062424795752683403288510
