| L(s) = 1 | + 5.20e3·9-s + 3.40e4·17-s + 3.55e5·25-s + 2.33e6·41-s − 1.64e6·49-s + 2.07e7·73-s + 1.55e7·81-s + 1.40e7·89-s + 7.86e6·97-s − 1.32e7·113-s + 3.41e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.77e8·153-s + 157-s + 163-s + 167-s + 3.63e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.38·9-s + 1.68·17-s + 4.54·25-s + 5.29·41-s − 1.99·49-s + 6.23·73-s + 3.25·81-s + 2.11·89-s + 0.875·97-s − 0.864·113-s + 1.75·121-s + 4.00·153-s + 0.579·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(29.49094762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.49094762\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - 868 p T^{2} + 88034 p^{3} T^{4} - 868 p^{15} T^{6} + p^{28} T^{8} )^{2} \) |
| 5 | \( ( 1 - 177556 T^{2} + 746525606 p^{2} T^{4} - 177556 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 7 | \( ( 1 + 820188 T^{2} + 26324522470 T^{4} + 820188 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 11 | \( ( 1 - 17075980 T^{2} + 829349603204918 T^{4} - 17075980 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 13 | \( ( 1 - 18167924 T^{2} + 2687884043393238 T^{4} - 18167924 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8508 T + 790404198 T^{2} - 8508 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 19 | \( ( 1 + 805600788 T^{2} + 1682716866294045334 T^{4} + 805600788 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10986212764 T^{2} + 52605343844079957158 T^{4} + 10986212764 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 29 | \( ( 1 - 40336999476 T^{2} + \)\(96\!\cdots\!70\)\( T^{4} - 40336999476 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 31 | \( ( 1 - 365644932 T^{2} - 25361902529689649786 T^{4} - 365644932 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 37 | \( ( 1 - 229993772052 T^{2} + \)\(25\!\cdots\!98\)\( T^{4} - 229993772052 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 41 | \( ( 1 - 584308 T + 451451590102 T^{2} - 584308 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 43 | \( ( 1 + 9686839668 T^{2} + \)\(13\!\cdots\!90\)\( T^{4} + 9686839668 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 47 | \( ( 1 + 1380372074812 T^{2} + \)\(97\!\cdots\!74\)\( T^{4} + 1380372074812 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 53 | \( ( 1 - 2723632239828 T^{2} + \)\(39\!\cdots\!18\)\( T^{4} - 2723632239828 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 59 | \( ( 1 - 5566423261772 T^{2} + \)\(17\!\cdots\!54\)\( T^{4} - 5566423261772 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 61 | \( ( 1 + 1358775604556 T^{2} + \)\(53\!\cdots\!50\)\( p^{2} T^{4} + 1358775604556 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 67 | \( ( 1 - 20239007070508 T^{2} + \)\(17\!\cdots\!50\)\( T^{4} - 20239007070508 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 71 | \( ( 1 + 11051333073628 T^{2} + \)\(13\!\cdots\!94\)\( T^{4} + 11051333073628 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 73 | \( ( 1 - 5184300 T + 27604353836630 T^{2} - 5184300 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 79 | \( ( 1 + 6129953969340 T^{2} - \)\(13\!\cdots\!42\)\( T^{4} + 6129953969340 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 83 | \( ( 1 - 36656479736044 T^{2} + \)\(52\!\cdots\!58\)\( T^{4} - 36656479736044 p^{14} T^{6} + p^{28} T^{8} )^{2} \) |
| 89 | \( ( 1 - 3521612 T + 77821818084598 T^{2} - 3521612 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
| 97 | \( ( 1 - 1967452 T + 160888317414278 T^{2} - 1967452 p^{7} T^{3} + p^{14} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.89643810086945123123598572082, −4.40241202937762825483525550735, −4.36808507403385095320453410417, −4.20438638979236700020684834322, −4.01128214895585891903849122781, −3.90313177438817436247859035157, −3.71090100053289682100597442413, −3.49575497540122431499633570388, −3.25112819750971553302906917889, −3.02730475273669681335157907421, −2.96509859138635011823092866510, −2.91710539144372594763154958344, −2.51211176998952565333101295706, −2.27571854941017606818228065314, −2.21005372534750850629938400325, −1.91464891747686678671211850804, −1.87704484417440617570117809459, −1.44447476083367936030102642783, −1.23482679370221672971642381432, −0.987548643019776401801926847366, −0.936479409722936666755910153696, −0.843664022616285010066573608080, −0.816635138311042436274772003707, −0.54096125394490792450807221097, −0.19107292056092898416158793139,
0.19107292056092898416158793139, 0.54096125394490792450807221097, 0.816635138311042436274772003707, 0.843664022616285010066573608080, 0.936479409722936666755910153696, 0.987548643019776401801926847366, 1.23482679370221672971642381432, 1.44447476083367936030102642783, 1.87704484417440617570117809459, 1.91464891747686678671211850804, 2.21005372534750850629938400325, 2.27571854941017606818228065314, 2.51211176998952565333101295706, 2.91710539144372594763154958344, 2.96509859138635011823092866510, 3.02730475273669681335157907421, 3.25112819750971553302906917889, 3.49575497540122431499633570388, 3.71090100053289682100597442413, 3.90313177438817436247859035157, 4.01128214895585891903849122781, 4.20438638979236700020684834322, 4.36808507403385095320453410417, 4.40241202937762825483525550735, 4.89643810086945123123598572082
Plot not available for L-functions of degree greater than 10.
