| L(s) = 1 | − 4·3-s − 2·4-s + 10·9-s + 8·12-s + 3·16-s − 16·17-s + 8·23-s − 2·25-s − 20·27-s + 8·29-s − 20·36-s + 20·43-s − 12·48-s + 20·49-s + 64·51-s − 24·53-s − 4·64-s + 32·68-s − 32·69-s + 8·75-s − 28·79-s + 35·81-s − 32·87-s − 16·92-s + 4·100-s − 16·101-s + 8·103-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s + 10/3·9-s + 2.30·12-s + 3/4·16-s − 3.88·17-s + 1.66·23-s − 2/5·25-s − 3.84·27-s + 1.48·29-s − 3.33·36-s + 3.04·43-s − 1.73·48-s + 20/7·49-s + 8.96·51-s − 3.29·53-s − 1/2·64-s + 3.88·68-s − 3.85·69-s + 0.923·75-s − 3.15·79-s + 35/9·81-s − 3.43·87-s − 1.66·92-s + 2/5·100-s − 1.59·101-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3469381488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3469381488\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T^{2} - 189 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1971 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 66 T^{2} + 3155 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 162 T^{2} + 12323 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 9270 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 9254 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 26006 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 37574 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 332 T^{2} + 45606 T^{4} - 332 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.83955675128917174586473787014, −5.62500485337379156825898025440, −5.49194736856076073513471977383, −5.18657772931539417043948211991, −4.85923643996565212245001478707, −4.83170884609190271036764818224, −4.73263198157037507525398455774, −4.51017453761159592094196942276, −4.27444606677148728188722586327, −4.17672741057560262241094813117, −4.16538192041527143214777956722, −3.82469716993044708573305040262, −3.40559991718207886799801950101, −3.35434524088864004605204738591, −2.83631891455693709046468565613, −2.73369351171195934949202708274, −2.47814259297317977755325093201, −2.20145882930819892389310735907, −2.10442867404915427308840673302, −1.60756357944892055932219947748, −1.21121071942937956481529495006, −1.21008786396763488387702735134, −0.799226027606984706254904247300, −0.42419887648281060949837645476, −0.18614522216857092229579863874,
0.18614522216857092229579863874, 0.42419887648281060949837645476, 0.799226027606984706254904247300, 1.21008786396763488387702735134, 1.21121071942937956481529495006, 1.60756357944892055932219947748, 2.10442867404915427308840673302, 2.20145882930819892389310735907, 2.47814259297317977755325093201, 2.73369351171195934949202708274, 2.83631891455693709046468565613, 3.35434524088864004605204738591, 3.40559991718207886799801950101, 3.82469716993044708573305040262, 4.16538192041527143214777956722, 4.17672741057560262241094813117, 4.27444606677148728188722586327, 4.51017453761159592094196942276, 4.73263198157037507525398455774, 4.83170884609190271036764818224, 4.85923643996565212245001478707, 5.18657772931539417043948211991, 5.49194736856076073513471977383, 5.62500485337379156825898025440, 5.83955675128917174586473787014
