| L(s) = 1 | + 2-s + 3·4-s + 9·8-s − 8·13-s + 3·16-s + 12·17-s − 8·26-s − 40·29-s + 17·32-s + 12·34-s + 128·37-s − 68·41-s + 60·49-s − 24·52-s − 152·53-s − 40·58-s − 112·61-s + 11·64-s + 36·68-s − 228·73-s + 128·74-s − 68·82-s + 220·89-s − 312·97-s + 60·98-s + 272·101-s − 72·104-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 3/4·4-s + 9/8·8-s − 0.615·13-s + 3/16·16-s + 0.705·17-s − 0.307·26-s − 1.37·29-s + 0.531·32-s + 6/17·34-s + 3.45·37-s − 1.65·41-s + 1.22·49-s − 0.461·52-s − 2.86·53-s − 0.689·58-s − 1.83·61-s + 0.171·64-s + 9/17·68-s − 3.12·73-s + 1.72·74-s − 0.829·82-s + 2.47·89-s − 3.21·97-s + 0.612·98-s + 2.69·101-s − 0.692·104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.509997821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.509997821\) |
| \(L(2)\) |
|
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 | $D_{4}$ | \( 1 - T - p T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2 \wr C_2$ | \( 1 - 60 T^{2} + 3078 T^{4} - 60 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 354 T^{2} + 56511 T^{4} - 354 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 178 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 423 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 74 T^{2} + 257911 T^{4} - 74 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1340 T^{2} + 1005958 T^{4} - 1340 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 20 T + 1618 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 3324 T^{2} + 4543686 T^{4} - 3324 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 64 T + 3598 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 4340 T^{2} + 9339718 T^{4} - 4340 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 3860 T^{2} + 12536998 T^{4} - 3860 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 76 T + 4438 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 244 T^{2} + 18936006 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 56 T + 4126 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 14370 T^{2} + 90627423 T^{4} - 14370 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 1044 T^{2} + 27414246 T^{4} - 1044 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 114 T + 13743 T^{2} + 114 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 17244 T^{2} + 144301446 T^{4} - 17244 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 22130 T^{2} + 213611503 T^{4} - 22130 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 110 T + 18703 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 156 T + 18998 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.94956248450852122830771182261, −6.88962447955149451900525292301, −6.64600482633740566305080849130, −6.28311384288515579485858362231, −6.06580291394607386245448545866, −5.79954014392390470344139557640, −5.70976014716519054722620890248, −5.69961273517289797896915677424, −5.02759868170215049984484565884, −4.80661269489482077122954505381, −4.72650258506101924630394185181, −4.42497738474182169563727867621, −4.34865360572490819256462679803, −4.04235386052048981092577224023, −3.45771128528020383723799119484, −3.41467726964923526430106853775, −3.06851447448135366014424506294, −2.93932574738720042521615050559, −2.47848577391031902234105276002, −2.16656889762977611528459256038, −1.90813792584659423438450132972, −1.46277156697651199723226867098, −1.42965873428830152250188481583, −0.56760585120768541649414412652, −0.47604766490144949924899155153,
0.47604766490144949924899155153, 0.56760585120768541649414412652, 1.42965873428830152250188481583, 1.46277156697651199723226867098, 1.90813792584659423438450132972, 2.16656889762977611528459256038, 2.47848577391031902234105276002, 2.93932574738720042521615050559, 3.06851447448135366014424506294, 3.41467726964923526430106853775, 3.45771128528020383723799119484, 4.04235386052048981092577224023, 4.34865360572490819256462679803, 4.42497738474182169563727867621, 4.72650258506101924630394185181, 4.80661269489482077122954505381, 5.02759868170215049984484565884, 5.69961273517289797896915677424, 5.70976014716519054722620890248, 5.79954014392390470344139557640, 6.06580291394607386245448545866, 6.28311384288515579485858362231, 6.64600482633740566305080849130, 6.88962447955149451900525292301, 6.94956248450852122830771182261
