| L(s) = 1 | − 4·5-s + 8·7-s + 92·11-s + 100·13-s − 92·17-s − 4·19-s + 8·23-s − 46·25-s − 84·29-s + 384·31-s − 32·35-s − 172·37-s + 300·41-s − 300·43-s − 16·47-s − 446·49-s + 12·53-s − 368·55-s − 644·59-s + 292·61-s − 400·65-s + 172·67-s + 408·71-s + 412·73-s + 736·77-s + 400·79-s + 948·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 0.431·7-s + 2.52·11-s + 2.13·13-s − 1.31·17-s − 0.0482·19-s + 0.0725·23-s − 0.367·25-s − 0.537·29-s + 2.22·31-s − 0.154·35-s − 0.764·37-s + 1.14·41-s − 1.06·43-s − 0.0496·47-s − 1.30·49-s + 0.0311·53-s − 0.902·55-s − 1.42·59-s + 0.612·61-s − 0.763·65-s + 0.313·67-s + 0.681·71-s + 0.660·73-s + 1.08·77-s + 0.569·79-s + 1.25·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.229998824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.229998824\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 510 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 92 T + 430 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 100 T + 5166 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 92 T + 5030 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 11370 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 45742 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 384 T + 84158 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 172 T + 99294 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 300 T + 157270 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 300 T + 180314 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 114782 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 288382 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 644 T + 462170 T^{2} + 644 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 292 T + 240078 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 172 T + 278250 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 408 T + 672766 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 412 T + 690678 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 400 T + 963870 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 948 T + 1360138 T^{2} - 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 572 T + 845846 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.573246124929127709405737165841, −8.954479129561972955900341278997, −8.764934620451859351775885757428, −8.711883612088204410309204229018, −7.900795239583206496533982475286, −7.80124671436334981273618982471, −6.98799612654727738299915886878, −6.45662512772631035007470417003, −6.29489710607757447726469706472, −6.27811325496867726685763215491, −5.37733307028289133126446210314, −4.69125893556177062952339285554, −4.36234374628858808470647334590, −4.00351961040418126977313039840, −3.37659229198499693330223410635, −3.27090056278619453963928088953, −1.92437585589779993798795365977, −1.86172203544796204926168754380, −0.983132755291107895015842329883, −0.68551433443494289931657579379,
0.68551433443494289931657579379, 0.983132755291107895015842329883, 1.86172203544796204926168754380, 1.92437585589779993798795365977, 3.27090056278619453963928088953, 3.37659229198499693330223410635, 4.00351961040418126977313039840, 4.36234374628858808470647334590, 4.69125893556177062952339285554, 5.37733307028289133126446210314, 6.27811325496867726685763215491, 6.29489710607757447726469706472, 6.45662512772631035007470417003, 6.98799612654727738299915886878, 7.80124671436334981273618982471, 7.900795239583206496533982475286, 8.711883612088204410309204229018, 8.764934620451859351775885757428, 8.954479129561972955900341278997, 9.573246124929127709405737165841
