| L(s) = 1 | + 144·11-s + 40·13-s − 432·23-s + 266·25-s − 400·37-s + 1.44e3·47-s + 718·49-s + 288·59-s + 752·61-s − 720·71-s − 1.00e3·73-s − 2.73e3·83-s + 2.09e3·97-s + 4.03e3·107-s − 4.68e3·109-s + 7.69e3·121-s + 127-s + 131-s + 137-s + 139-s + 5.76e3·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + ⋯ |
| L(s) = 1 | + 3.94·11-s + 0.853·13-s − 3.91·23-s + 2.12·25-s − 1.77·37-s + 4.46·47-s + 2.09·49-s + 0.635·59-s + 1.57·61-s − 1.20·71-s − 1.60·73-s − 3.61·83-s + 2.18·97-s + 3.64·107-s − 4.11·109-s + 5.77·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 3.36·143-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.00546·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.249588630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.249588630\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 266 T^{2} + 45051 T^{4} - 266 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 718 T^{2} + 360291 T^{4} - 718 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 72 T + 3931 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 20 T + 606 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12668 T^{2} + 76201926 T^{4} - 12668 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 24460 T^{2} + 242669334 T^{4} - 24460 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 216 T + 32110 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 14540 T^{2} + 285989334 T^{4} - 14540 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 102478 T^{2} + 4392569283 T^{4} - 102478 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 200 T + 76314 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4252 T^{2} - 5188552410 T^{4} + 4252 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 142420 T^{2} + 11541001398 T^{4} - 142420 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 720 T + 324178 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 17014 T^{2} + 16994482107 T^{4} + 17014 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 144 T + 156634 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 376 T + 427098 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1096180 T^{2} + 479370498006 T^{4} - 1096180 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 360 T + 717010 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 502 T + 370587 T^{2} + 502 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 86492 T^{2} - 48924623994 T^{4} + 86492 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1368 T + 1059307 T^{2} + 1368 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1783868 T^{2} + 1521664522278 T^{4} - 1783868 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1046 T + 2036667 T^{2} - 1046 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.45069433820372933408140011325, −7.14317884364119441014960310117, −7.13122378571049246141410973309, −7.09824246184342046795486451092, −6.58186570410277785892047426363, −6.34786841723425157622980565996, −6.11651528586613948859570846355, −5.95094078554756094323165281661, −5.70064824001362034633836133367, −5.57370510739316837923240940092, −5.10093633793027232343881298444, −4.59094720028801526341671117240, −4.28545963183724762223200672371, −4.11075004898636461526355859674, −3.91818132539597465826074870813, −3.80233077029571880167730659434, −3.62631214545437935937831509828, −2.99088620039862275517369946625, −2.61600213270641825445243126782, −2.30039482257340555406620674958, −1.65794004840322212209700559744, −1.63223349127180174178892047862, −1.25798179337557768545434699364, −0.62534123095145325551838549072, −0.60402484861914518125967116382,
0.60402484861914518125967116382, 0.62534123095145325551838549072, 1.25798179337557768545434699364, 1.63223349127180174178892047862, 1.65794004840322212209700559744, 2.30039482257340555406620674958, 2.61600213270641825445243126782, 2.99088620039862275517369946625, 3.62631214545437935937831509828, 3.80233077029571880167730659434, 3.91818132539597465826074870813, 4.11075004898636461526355859674, 4.28545963183724762223200672371, 4.59094720028801526341671117240, 5.10093633793027232343881298444, 5.57370510739316837923240940092, 5.70064824001362034633836133367, 5.95094078554756094323165281661, 6.11651528586613948859570846355, 6.34786841723425157622980565996, 6.58186570410277785892047426363, 7.09824246184342046795486451092, 7.13122378571049246141410973309, 7.14317884364119441014960310117, 7.45069433820372933408140011325
