| L(s) = 1 | + 112·13-s + 8·19-s − 38·25-s − 20·31-s + 160·37-s − 80·43-s − 8·61-s − 144·67-s − 288·73-s + 140·79-s + 552·97-s − 232·103-s − 192·109-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.88e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 8.61·13-s + 8/19·19-s − 1.51·25-s − 0.645·31-s + 4.32·37-s − 1.86·43-s − 0.131·61-s − 2.14·67-s − 3.94·73-s + 1.77·79-s + 5.69·97-s − 2.25·103-s − 1.76·109-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 34.8·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(18.38202467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.38202467\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 38 T^{2} + 569 T^{4} - 114 p^{3} T^{6} - 844 p^{4} T^{8} - 114 p^{7} T^{10} + 569 p^{8} T^{12} + 38 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 2 p^{2} T^{2} + 23657 T^{4} + 11250 p^{2} T^{6} + 146704244 T^{8} + 11250 p^{6} T^{10} + 23657 p^{8} T^{12} + 2 p^{14} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 28 T + 488 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( 1 + 320 T^{2} - 16642 T^{4} - 15360000 T^{6} - 1357310077 T^{8} - 15360000 p^{4} T^{10} - 16642 p^{8} T^{12} + 320 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 4 T - 296 T^{2} + 1640 T^{3} - 38753 T^{4} + 1640 p^{2} T^{5} - 296 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 832 T^{2} - 28738 T^{4} - 134184960 T^{6} + 247762563779 T^{8} - 134184960 p^{4} T^{10} - 28738 p^{8} T^{12} - 832 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 3302 T^{2} + 4139627 T^{4} - 3302 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 10 T - 1801 T^{2} - 210 T^{3} + 2594180 T^{4} - 210 p^{2} T^{5} - 1801 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 80 T + 2476 T^{2} - 94880 T^{3} + 4762015 T^{4} - 94880 p^{2} T^{5} + 2476 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 4492 T^{2} + 9742182 T^{4} - 4492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 20 T + 3384 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 2960 T^{2} + 3881919 T^{4} + 2960 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 + 10678 T^{2} + 69793417 T^{4} + 303738966790 T^{6} + 994032852115924 T^{8} + 303738966790 p^{4} T^{10} + 69793417 p^{8} T^{12} + 10678 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 430 T^{2} + 3177353 T^{4} + 11707685250 T^{6} - 140116162696012 T^{8} + 11707685250 p^{4} T^{10} + 3177353 p^{8} T^{12} - 430 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 4 T - 5176 T^{2} - 9000 T^{3} + 13051487 T^{4} - 9000 p^{2} T^{5} - 5176 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 72 T - 4354 T^{2} + 40320 T^{3} + 47551347 T^{4} + 40320 p^{2} T^{5} - 4354 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12032 T^{2} + 81572162 T^{4} - 12032 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 18 T - 35 p T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 162 T + 205 p T^{2} + 162 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | \( ( 1 - 70 T - 8761 T^{2} - 82530 T^{3} + 117091940 T^{4} - 82530 p^{2} T^{5} - 8761 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 5858 T^{2} - 4749493 T^{4} - 5858 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 + 13180 T^{2} + 58454218 T^{4} - 134782634000 T^{6} - 1551245553245357 T^{8} - 134782634000 p^{4} T^{10} + 58454218 p^{8} T^{12} + 13180 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 - 138 T + 11803 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−3.83565895020414910621334602557, −3.45410596975797331862851422408, −3.43400910572360288491306209821, −3.33719500578873574360134649739, −3.26728995801569193413241081049, −3.23051696262613610417394497963, −3.03400200695606223979205688692, −2.96446058034307030404032261872, −2.73290863751949844274630042653, −2.63999990879053494633434559795, −2.44920363308145973381313952000, −2.09082279590223804372154456982, −2.05438406090068790183962572040, −1.87492230495744215666002077019, −1.82304767453840287941336702842, −1.59862380903962026508740446226, −1.42070951255444073721862219833, −1.36084919712681912641688969989, −1.24694124946524719149034360797, −1.05788605634157605063398891569, −0.908588184103309211267015565575, −0.820256116128002485414803292358, −0.77564595431379262809613104991, −0.30542596914294687256076729350, −0.18902310952873582951338031805,
0.18902310952873582951338031805, 0.30542596914294687256076729350, 0.77564595431379262809613104991, 0.820256116128002485414803292358, 0.908588184103309211267015565575, 1.05788605634157605063398891569, 1.24694124946524719149034360797, 1.36084919712681912641688969989, 1.42070951255444073721862219833, 1.59862380903962026508740446226, 1.82304767453840287941336702842, 1.87492230495744215666002077019, 2.05438406090068790183962572040, 2.09082279590223804372154456982, 2.44920363308145973381313952000, 2.63999990879053494633434559795, 2.73290863751949844274630042653, 2.96446058034307030404032261872, 3.03400200695606223979205688692, 3.23051696262613610417394497963, 3.26728995801569193413241081049, 3.33719500578873574360134649739, 3.43400910572360288491306209821, 3.45410596975797331862851422408, 3.83565895020414910621334602557
Plot not available for L-functions of degree greater than 10.
