| L(s) = 1 | − 4·4-s − 12·13-s + 12·16-s − 60·19-s + 16·31-s + 112·37-s − 220·43-s + 14·49-s + 48·52-s − 88·61-s − 32·64-s − 152·67-s + 84·73-s + 240·76-s + 144·79-s + 152·97-s + 372·103-s − 208·109-s + 20·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s − 448·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 0.923·13-s + 3/4·16-s − 3.15·19-s + 0.516·31-s + 3.02·37-s − 5.11·43-s + 2/7·49-s + 0.923·52-s − 1.44·61-s − 1/2·64-s − 2.26·67-s + 1.15·73-s + 3.15·76-s + 1.82·79-s + 1.56·97-s + 3.61·103-s − 1.90·109-s + 0.165·121-s − 0.516·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 3.02·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7461820654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7461820654\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 12121 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 6 T + 340 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 112 T^{2} + 40706 T^{4} + 112 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 30 T + 772 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 932 T^{2} + 775655 T^{4} - 932 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2492 T^{2} + 2785631 T^{4} - 2492 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1490 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 56 T + 3347 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4420 T^{2} + 10453974 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 110 T + 6471 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1960 T^{2} + 7090962 T^{4} - 1960 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3632 T^{2} + 7466210 T^{4} - 3632 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3952 T^{2} + 19974498 T^{4} - 3952 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 44 T + 3894 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 76 T + 10079 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6260 T^{2} + 48182039 T^{4} - 6260 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 42 T + 2524 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 72 T + 11251 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 11900 T^{2} + 120439734 T^{4} + 11900 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 11048 T^{2} + 54493010 T^{4} - 11048 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 76 T + 11190 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.04854582812301759919637820897, −5.85977464889104087804259113717, −5.57328944861927304664254719721, −5.17978637618851895787526468988, −5.07545339769322211368103899467, −4.81459431158422862767917126337, −4.76082404994267273312198742763, −4.62462177208030621617084561184, −4.28851928612754953852517393893, −4.25785581835617856839318231187, −4.03318313222066580096916384070, −3.61050061596892274450918360000, −3.40317478616610154073416093323, −3.31761586976806187637685049435, −3.10166195969765012412629430577, −2.66644401460672519472926304172, −2.39293798896131396660726317228, −2.37166323785618660372081003705, −1.88833898468521506323046117554, −1.78163119195540592764391620081, −1.59296067695501956559508679417, −0.961963074790999388777519628256, −0.874997305559815103820632212731, −0.28664570175767838873006466345, −0.20719098661605582395711551994,
0.20719098661605582395711551994, 0.28664570175767838873006466345, 0.874997305559815103820632212731, 0.961963074790999388777519628256, 1.59296067695501956559508679417, 1.78163119195540592764391620081, 1.88833898468521506323046117554, 2.37166323785618660372081003705, 2.39293798896131396660726317228, 2.66644401460672519472926304172, 3.10166195969765012412629430577, 3.31761586976806187637685049435, 3.40317478616610154073416093323, 3.61050061596892274450918360000, 4.03318313222066580096916384070, 4.25785581835617856839318231187, 4.28851928612754953852517393893, 4.62462177208030621617084561184, 4.76082404994267273312198742763, 4.81459431158422862767917126337, 5.07545339769322211368103899467, 5.17978637618851895787526468988, 5.57328944861927304664254719721, 5.85977464889104087804259113717, 6.04854582812301759919637820897
