| L(s) = 1 | + 16·7-s + 8·11-s − 16·17-s + 48·23-s + 124·31-s − 32·37-s + 112·41-s + 112·43-s + 160·47-s + 128·49-s − 208·53-s + 300·61-s − 144·67-s + 272·71-s − 224·73-s + 128·77-s − 9·81-s + 160·83-s + 320·97-s + 224·101-s − 96·103-s + 144·107-s + 320·113-s − 256·119-s − 252·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 16/7·7-s + 8/11·11-s − 0.941·17-s + 2.08·23-s + 4·31-s − 0.864·37-s + 2.73·41-s + 2.60·43-s + 3.40·47-s + 2.61·49-s − 3.92·53-s + 4.91·61-s − 2.14·67-s + 3.83·71-s − 3.06·73-s + 1.66·77-s − 1/9·81-s + 1.92·83-s + 3.29·97-s + 2.21·101-s − 0.932·103-s + 1.34·107-s + 2.83·113-s − 2.15·119-s − 2.08·121-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(16.55676578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.55676578\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 864 T^{3} + 5807 T^{4} - 864 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 39599 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 3408 T^{3} + 84962 T^{4} + 3408 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 286 T^{2} + 277635 T^{4} + 286 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 1680 p T^{3} + 2306 p^{2} T^{4} - 1680 p^{3} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1244 T^{2} + 1124070 T^{4} - 1244 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 p T + 2787 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 46368 T^{3} + 4192802 T^{4} + 46368 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 56 T + 4050 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T + 6272 T^{2} - 366240 T^{3} + 19366559 T^{4} - 366240 p^{2} T^{5} + 6272 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} - 817440 T^{3} + 43793762 T^{4} - 817440 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 208 T + 21632 T^{2} + 1699152 T^{3} + 104735522 T^{4} + 1699152 p^{2} T^{5} + 21632 p^{4} T^{6} + 208 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 150 T + 12683 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 144 T + 10368 T^{2} + 863712 T^{3} + 69674927 T^{4} + 863712 p^{2} T^{5} + 10368 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 136 T + 14610 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 224 T + 25088 T^{2} + 2071776 T^{3} + 155721602 T^{4} + 2071776 p^{2} T^{5} + 25088 p^{4} T^{6} + 224 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 5692 T^{2} + 84617478 T^{4} - 5692 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} - 1520160 T^{3} + 173715458 T^{4} - 1520160 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8284 T^{2} + 112535046 T^{4} + 8284 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 320 T + 51200 T^{2} - 6760320 T^{3} + 755327663 T^{4} - 6760320 p^{2} T^{5} + 51200 p^{4} T^{6} - 320 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.80189612705877283534397818584, −6.35587791399098896787949532179, −6.24023023809292564239193458722, −6.18100292542559482469094119473, −6.14960552992569689859605442717, −5.52135445286301082834356142515, −5.25436165036713408521586313285, −5.12927428276685440828545320374, −5.10442916701217774549053309985, −4.53050767150003900336471048473, −4.44694102277477341783542257109, −4.33199286530046007212112688156, −4.30923680246525737189581120429, −3.80434058044551649044685991543, −3.54631896612107099106035003410, −3.07484698004311830843967525652, −2.95416454841826245865020388031, −2.43961661149340519364388963937, −2.32634213244644553718264783877, −2.21212442770612017157979327643, −1.82733288391651263061335503636, −1.04161757480376773267314324728, −0.999477736977869998528879247165, −0.991774954799724129971549903462, −0.59213401577003809036430115903,
0.59213401577003809036430115903, 0.991774954799724129971549903462, 0.999477736977869998528879247165, 1.04161757480376773267314324728, 1.82733288391651263061335503636, 2.21212442770612017157979327643, 2.32634213244644553718264783877, 2.43961661149340519364388963937, 2.95416454841826245865020388031, 3.07484698004311830843967525652, 3.54631896612107099106035003410, 3.80434058044551649044685991543, 4.30923680246525737189581120429, 4.33199286530046007212112688156, 4.44694102277477341783542257109, 4.53050767150003900336471048473, 5.10442916701217774549053309985, 5.12927428276685440828545320374, 5.25436165036713408521586313285, 5.52135445286301082834356142515, 6.14960552992569689859605442717, 6.18100292542559482469094119473, 6.24023023809292564239193458722, 6.35587791399098896787949532179, 6.80189612705877283534397818584
