L(s) = 1 | − 82·5-s − 98·7-s − 340·11-s + 910·13-s − 3.21e3·17-s − 674·19-s + 1.10e3·23-s + 1.89e3·25-s − 8.06e3·29-s − 6.21e3·31-s + 8.03e3·35-s − 8.51e3·37-s + 1.30e3·41-s − 1.00e4·43-s + 1.27e4·47-s + 7.20e3·49-s + 1.12e4·53-s + 2.78e4·55-s + 1.20e4·59-s + 1.02e5·61-s − 7.46e4·65-s + 2.41e4·67-s − 8.97e4·71-s − 5.55e4·73-s + 3.33e4·77-s + 4.88e4·79-s − 3.57e4·83-s + ⋯ |
L(s) = 1 | − 1.46·5-s − 0.755·7-s − 0.847·11-s + 1.49·13-s − 2.69·17-s − 0.428·19-s + 0.435·23-s + 0.607·25-s − 1.78·29-s − 1.16·31-s + 1.10·35-s − 1.02·37-s + 0.121·41-s − 0.825·43-s + 0.841·47-s + 3/7·49-s + 0.548·53-s + 1.24·55-s + 0.449·59-s + 3.53·61-s − 2.19·65-s + 0.656·67-s − 2.11·71-s − 1.22·73-s + 0.640·77-s + 0.880·79-s − 0.570·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2949645760\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2949645760\) |
\(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
good | 5 | $D_{4}$ | \( 1 + 82 T + 4826 T^{2} + 82 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 340 T + 338582 T^{2} + 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 70 p T + 2514 p^{2} T^{2} - 70 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3216 T + 5412958 T^{2} + 3216 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 674 T + 4367142 T^{2} + 674 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 48 p T + 495170 p T^{2} - 48 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8064 T + 52795702 T^{2} + 8064 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6212 T + 51691038 T^{2} + 6212 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8512 T + 104326950 T^{2} + 8512 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1304 T + 73546526 T^{2} - 1304 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10004 T + 99339510 T^{2} + 10004 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12748 T + 323438270 T^{2} - 12748 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11220 T + 664373806 T^{2} - 11220 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12018 T - 285266426 T^{2} - 12018 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 102738 T + 4326026138 T^{2} - 102738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24136 T + 1542084918 T^{2} - 24136 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 89720 T + 4576356302 T^{2} + 89720 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 55588 T + 3902743302 T^{2} + 55588 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 48824 T + 5430110622 T^{2} - 48824 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 35782 T + 4098945062 T^{2} + 35782 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18300 T + 3539716918 T^{2} - 18300 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 69984 T + 18325482398 T^{2} + 69984 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.29760562232428914653446226239, −10.18521691518459980932659029664, −9.088181542718233345962461923164, −9.032118767071783938773622721307, −8.560199287286132480420542892610, −8.299608938060414423788276387688, −7.40496902281153019880169429420, −7.33472597959012779974304650735, −6.74338739828934985247254892415, −6.34927608121682947950674116515, −5.64415260502164400458414142314, −5.28374601000198431497692382319, −4.29002479174490420073573869110, −4.23884852699875215739235091594, −3.48144762876324781052970331515, −3.33560127601446738634115037361, −2.16838071857874808751249940652, −2.02311823499215566510609945106, −0.75524498179294752771609968315, −0.17186420559463848034442710233,
0.17186420559463848034442710233, 0.75524498179294752771609968315, 2.02311823499215566510609945106, 2.16838071857874808751249940652, 3.33560127601446738634115037361, 3.48144762876324781052970331515, 4.23884852699875215739235091594, 4.29002479174490420073573869110, 5.28374601000198431497692382319, 5.64415260502164400458414142314, 6.34927608121682947950674116515, 6.74338739828934985247254892415, 7.33472597959012779974304650735, 7.40496902281153019880169429420, 8.299608938060414423788276387688, 8.560199287286132480420542892610, 9.032118767071783938773622721307, 9.088181542718233345962461923164, 10.18521691518459980932659029664, 10.29760562232428914653446226239