| L(s) = 1 | + 4·2-s + 8·4-s − 16·7-s + 8·8-s − 16·11-s + 12·13-s − 64·14-s − 4·16-s + 44·17-s − 64·22-s − 16·23-s + 46·25-s + 48·26-s − 128·28-s − 80·31-s − 32·32-s + 176·34-s − 108·37-s + 32·41-s + 48·43-s − 128·44-s − 64·46-s + 96·47-s + 128·49-s + 184·50-s + 96·52-s + 100·53-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s − 2.28·7-s + 8-s − 1.45·11-s + 0.923·13-s − 4.57·14-s − 1/4·16-s + 2.58·17-s − 2.90·22-s − 0.695·23-s + 1.83·25-s + 1.84·26-s − 4.57·28-s − 2.58·31-s − 32-s + 5.17·34-s − 2.91·37-s + 0.780·41-s + 1.11·43-s − 2.90·44-s − 1.39·46-s + 2.04·47-s + 2.61·49-s + 3.67·50-s + 1.84·52-s + 1.88·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.855336862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855336862\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 528 T^{3} + 1922 T^{4} + 528 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 162 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 60 T^{3} - 30226 T^{4} + 60 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T + 968 T^{2} - 21252 T^{3} + 428942 T^{4} - 21252 p^{2} T^{5} + 968 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 740 T^{2} + 299238 T^{4} - 740 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 2064 T^{3} - 126718 T^{4} + 2064 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2660 T^{2} + 3085158 T^{4} - 2660 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 40 T + 1938 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 44976 T^{3} + 924194 T^{4} - 44976 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 96 T + 4608 T^{2} - 281184 T^{3} + 16639682 T^{4} - 281184 p^{2} T^{5} + 4608 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T + 5000 T^{2} - 17100 T^{3} - 6900562 T^{4} - 17100 p^{2} T^{5} + 5000 p^{4} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 48 T + 6482 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 10128 T^{3} - 14067358 T^{4} + 10128 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 32 T + 10242 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 188 T + 17672 T^{2} + 1796340 T^{3} + 164736974 T^{4} + 1796340 p^{2} T^{5} + 17672 p^{4} T^{6} + 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7772 T^{2} + 83656134 T^{4} + 7772 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 192 T + 18432 T^{2} - 1755840 T^{3} + 162172514 T^{4} - 1755840 p^{2} T^{5} + 18432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14468 T^{2} + 151051974 T^{4} - 14468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 132 T + 8712 T^{2} - 895884 T^{3} + 85251854 T^{4} - 895884 p^{2} T^{5} + 8712 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−12.75942140833946660143196755059, −12.34404933195726100747417669697, −12.27079893407829307417435633175, −12.03573357772855083898470178735, −11.57826483071142711382839765029, −10.67551265776307764616745642546, −10.67401117978856701201658190550, −10.37146705466737872868219592587, −10.28518989466324697775697227981, −9.394821779975947401099552855393, −9.187368065725560403528322552354, −8.978311315411257178099739521632, −8.349193146727985219638768480258, −7.69177153425512979347890941581, −7.33458185583414642348968177730, −7.04669331068887754683718107643, −6.54895168565730344583305656571, −5.85391794536876729071588751358, −5.66628190222639169842756353527, −5.53373706520711472499091565625, −4.91155903697854065693088417028, −3.95063285925117270929903707532, −3.56704809263669712972749869582, −3.26110301690589559140612416868, −2.65158771136712994184584443690,
2.65158771136712994184584443690, 3.26110301690589559140612416868, 3.56704809263669712972749869582, 3.95063285925117270929903707532, 4.91155903697854065693088417028, 5.53373706520711472499091565625, 5.66628190222639169842756353527, 5.85391794536876729071588751358, 6.54895168565730344583305656571, 7.04669331068887754683718107643, 7.33458185583414642348968177730, 7.69177153425512979347890941581, 8.349193146727985219638768480258, 8.978311315411257178099739521632, 9.187368065725560403528322552354, 9.394821779975947401099552855393, 10.28518989466324697775697227981, 10.37146705466737872868219592587, 10.67401117978856701201658190550, 10.67551265776307764616745642546, 11.57826483071142711382839765029, 12.03573357772855083898470178735, 12.27079893407829307417435633175, 12.34404933195726100747417669697, 12.75942140833946660143196755059
