| L(s) = 1 | + 1.15e4·3-s − 7.91e5·5-s − 1.89e7·7-s − 6.20e6·9-s − 4.70e8·11-s − 2.50e9·13-s − 9.17e9·15-s − 4.84e10·17-s − 8.24e10·19-s − 2.19e11·21-s − 3.07e11·23-s − 3.55e11·25-s − 2.04e11·27-s + 1.98e12·29-s + 1.07e13·31-s − 5.45e12·33-s + 1.49e13·35-s − 5.11e13·37-s − 2.90e13·39-s − 1.14e14·41-s − 5.67e13·43-s + 4.91e12·45-s + 2.01e14·47-s + 1.82e14·49-s − 5.61e14·51-s + 2.17e14·53-s + 3.72e14·55-s + ⋯ |
| L(s) = 1 | + 1.02·3-s − 0.906·5-s − 1.24·7-s − 0.0480·9-s − 0.662·11-s − 0.851·13-s − 0.924·15-s − 1.68·17-s − 1.11·19-s − 1.26·21-s − 0.817·23-s − 0.466·25-s − 0.139·27-s + 0.738·29-s + 2.26·31-s − 0.675·33-s + 1.12·35-s − 2.39·37-s − 0.868·39-s − 2.23·41-s − 0.740·43-s + 0.0435·45-s + 1.23·47-s + 0.783·49-s − 1.71·51-s + 0.479·53-s + 0.600·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
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 | $D_{4}$ | \( 1 - 1288 p^{2} T + 1735606 p^{4} T^{2} - 1288 p^{19} T^{3} + p^{34} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 791924 T + 39324020702 p^{2} T^{2} + 791924 p^{17} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2704656 p T + 3596633354414 p^{2} T^{2} + 2704656 p^{18} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 470771240 T + 95486052235529266 p T^{2} + 470771240 p^{17} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2503523428 T + 1422330837350947926 p T^{2} + 2503523428 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 48444688348 T + \)\(19\!\cdots\!54\)\( T^{2} + 48444688348 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 82483300760 T + \)\(67\!\cdots\!42\)\( T^{2} + 82483300760 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 307097031248 T + \)\(29\!\cdots\!78\)\( T^{2} + 307097031248 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1989575387580 T + \)\(15\!\cdots\!74\)\( T^{2} - 1989575387580 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10752133575232 T + \)\(63\!\cdots\!62\)\( T^{2} - 10752133575232 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 51174452749620 T + \)\(15\!\cdots\!18\)\( T^{2} + 51174452749620 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 114227291044524 T + \)\(71\!\cdots\!10\)\( T^{2} + 114227291044524 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 56757554203624 T + \)\(98\!\cdots\!74\)\( T^{2} + 56757554203624 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 201553163158368 T + \)\(47\!\cdots\!30\)\( T^{2} - 201553163158368 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 217275463587052 T - \)\(60\!\cdots\!98\)\( T^{2} - 217275463587052 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 466311448890056 T + \)\(19\!\cdots\!86\)\( T^{2} + 466311448890056 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2063817263626748 T + \)\(39\!\cdots\!02\)\( T^{2} - 2063817263626748 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6637225962358328 T + \)\(48\!\cdots\!18\)\( p T^{2} + 6637225962358328 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1543684997984528 T + \)\(56\!\cdots\!22\)\( p T^{2} - 1543684997984528 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3734106800358508 T + \)\(85\!\cdots\!38\)\( T^{2} + 3734106800358508 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 17755378507330976 T + \)\(39\!\cdots\!58\)\( T^{2} - 17755378507330976 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25453597558158056 T + \)\(65\!\cdots\!54\)\( T^{2} - 25453597558158056 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 67216838659203660 T + \)\(33\!\cdots\!82\)\( T^{2} + 67216838659203660 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 130979024861443388 T + \)\(10\!\cdots\!10\)\( T^{2} + 130979024861443388 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−17.07124668846459616164963814137, −16.02303527477478068065083943562, −15.38976912986250374899757385059, −15.17175385439891435008896671574, −13.66785306438138202927343460632, −13.64591992132319105430419518609, −12.36666481168133633811470448960, −11.87228890542891887047418456211, −10.50288632577951408328928562885, −9.907002773695105674602431999129, −8.533537834164092254463316946090, −8.463905006346942247990001219794, −7.11319519906441161480164699959, −6.38094328894666818679206471059, −4.78370089346651827781237397782, −3.79657498476383608821472282402, −2.86785751013112843534804536545, −2.17958202452771698637039419912, 0, 0,
2.17958202452771698637039419912, 2.86785751013112843534804536545, 3.79657498476383608821472282402, 4.78370089346651827781237397782, 6.38094328894666818679206471059, 7.11319519906441161480164699959, 8.463905006346942247990001219794, 8.533537834164092254463316946090, 9.907002773695105674602431999129, 10.50288632577951408328928562885, 11.87228890542891887047418456211, 12.36666481168133633811470448960, 13.64591992132319105430419518609, 13.66785306438138202927343460632, 15.17175385439891435008896671574, 15.38976912986250374899757385059, 16.02303527477478068065083943562, 17.07124668846459616164963814137
