| L(s) = 1 | + 12·5-s − 6·9-s − 36·17-s + 64·25-s + 48·29-s + 64·37-s + 132·41-s − 72·45-s + 192·53-s + 264·61-s − 24·73-s + 27·81-s − 432·85-s + 276·89-s + 600·97-s − 252·101-s − 280·109-s − 168·113-s + 256·121-s + 372·125-s + 127-s + 131-s + 137-s + 139-s + 576·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 12/5·5-s − 2/3·9-s − 2.11·17-s + 2.55·25-s + 1.65·29-s + 1.72·37-s + 3.21·41-s − 8/5·45-s + 3.62·53-s + 4.32·61-s − 0.328·73-s + 1/3·81-s − 5.08·85-s + 3.10·89-s + 6.18·97-s − 2.49·101-s − 2.56·109-s − 1.48·113-s + 2.11·121-s + 2.97·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.97·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(18.52378418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.52378418\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 6 T + 22 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 44334 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $D_{4}$ | \( ( 1 + 18 T + 622 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 p T^{2} + 310086 T^{4} - 28 p^{5} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1168 T^{2} + 739566 T^{4} - 1168 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 494 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 p T^{2} + 2701926 T^{4} - 76 p^{5} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 32 T + 1662 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 66 T + 4414 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2930 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4420 T^{2} + 11574534 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 96 T + 6590 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5186 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 132 T + 10466 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9748 T^{2} + 62331846 T^{4} - 9748 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 14320 T^{2} + 100457262 T^{4} - 14320 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 9362 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1972 T^{2} - 41007642 T^{4} - 1972 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20548 T^{2} + 188196006 T^{4} - 20548 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 138 T + 14350 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 300 T + 39986 T^{2} - 300 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.31976577567472739503370425911, −5.83628324918865413870093828695, −5.74376733350372437952058023843, −5.66616781807950182003719074407, −5.51543927383642575911811987539, −5.27215587215522862712307775107, −4.87603016350702272286427281518, −4.76465153890887823826987850357, −4.62032153509005319565462432694, −4.29782674744180591255982807253, −4.01196415197555433480959668295, −3.93725313556305781825995639281, −3.67754658176515912515547159333, −3.33760702428357902226630605353, −2.73042008745469381300123129485, −2.70222769996513676642567885309, −2.67876800519266624528750729045, −2.31215675535558718170582129274, −2.14448872057675879423244490840, −1.92730300080200392340613779413, −1.81098434828358126156703514361, −0.969271288031785980815988586064, −0.959590973427468262272272259386, −0.60859400626210769332674704209, −0.55929956857743725713379986441,
0.55929956857743725713379986441, 0.60859400626210769332674704209, 0.959590973427468262272272259386, 0.969271288031785980815988586064, 1.81098434828358126156703514361, 1.92730300080200392340613779413, 2.14448872057675879423244490840, 2.31215675535558718170582129274, 2.67876800519266624528750729045, 2.70222769996513676642567885309, 2.73042008745469381300123129485, 3.33760702428357902226630605353, 3.67754658176515912515547159333, 3.93725313556305781825995639281, 4.01196415197555433480959668295, 4.29782674744180591255982807253, 4.62032153509005319565462432694, 4.76465153890887823826987850357, 4.87603016350702272286427281518, 5.27215587215522862712307775107, 5.51543927383642575911811987539, 5.66616781807950182003719074407, 5.74376733350372437952058023843, 5.83628324918865413870093828695, 6.31976577567472739503370425911
