| L(s) = 1 | + 3·2-s + 5·4-s − 10·7-s + 27·8-s + 42·11-s − 64·13-s − 30·14-s + 69·16-s + 18·17-s + 130·19-s + 126·22-s + 114·23-s − 192·26-s − 50·28-s + 240·29-s − 8·31-s + 27·32-s + 54·34-s − 184·37-s + 390·38-s − 672·41-s − 370·43-s + 210·44-s + 342·46-s − 276·47-s − 98·49-s − 320·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 5/8·4-s − 0.539·7-s + 1.19·8-s + 1.15·11-s − 1.36·13-s − 0.572·14-s + 1.07·16-s + 0.256·17-s + 1.56·19-s + 1.22·22-s + 1.03·23-s − 1.44·26-s − 0.337·28-s + 1.53·29-s − 0.0463·31-s + 0.149·32-s + 0.272·34-s − 0.817·37-s + 1.66·38-s − 2.55·41-s − 1.31·43-s + 0.719·44-s + 1.09·46-s − 0.856·47-s − 2/7·49-s − 0.853·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.295431802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.295431802\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{2} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 198 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 42 T + 3046 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 64 T + 4905 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 18 T + 1699 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 130 T + 13326 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 114 T + 20686 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 240 T + 53545 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 26766 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 184 T + 105153 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 672 T + 239566 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 370 T + 4482 p T^{2} + 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 276 T + 226462 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 204 T + 300550 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 208 T + 193401 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 802 T + 675630 T^{2} + 802 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 126 T + 534598 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 374 T + 646791 T^{2} - 374 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1378 T + 1460286 T^{2} - 1378 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1164 T + 1339798 T^{2} - 1164 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 954 T + 816667 T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 872 T + 1964142 T^{2} - 872 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.063092873262496692475019714274, −8.569251609786680114024607780252, −8.085015178421102641522569469595, −7.75247411874144122379342016536, −7.24369361840146110128671741654, −6.79949486803704303058593079260, −6.74505589170069873367920780470, −6.39190331616703389462464936454, −5.57574655866817147420910222900, −5.20178940881510309539975765080, −4.85711008367053715685514991473, −4.83124468858562309444154868804, −4.00620300696818916470407866904, −3.65476248426657754007129212714, −3.13566621227458585058162403107, −2.96375898578411664562767134112, −2.16318191278434175318393542328, −1.52953136568009304547077347859, −1.21386162102165379302201800999, −0.39237977728001835308792477104,
0.39237977728001835308792477104, 1.21386162102165379302201800999, 1.52953136568009304547077347859, 2.16318191278434175318393542328, 2.96375898578411664562767134112, 3.13566621227458585058162403107, 3.65476248426657754007129212714, 4.00620300696818916470407866904, 4.83124468858562309444154868804, 4.85711008367053715685514991473, 5.20178940881510309539975765080, 5.57574655866817147420910222900, 6.39190331616703389462464936454, 6.74505589170069873367920780470, 6.79949486803704303058593079260, 7.24369361840146110128671741654, 7.75247411874144122379342016536, 8.085015178421102641522569469595, 8.569251609786680114024607780252, 9.063092873262496692475019714274
