| L(s) = 1 | − 8·7-s + 16·13-s + 48·31-s + 16·43-s + 32·49-s − 48·61-s + 48·67-s − 40·73-s − 128·91-s + 8·97-s − 8·103-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 4.43·13-s + 8.62·31-s + 2.43·43-s + 32/7·49-s − 6.14·61-s + 5.86·67-s − 4.68·73-s − 13.4·91-s + 0.812·97-s − 0.788·103-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.492459701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.492459701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 4 T + 8 T^{2} - 4 T^{3} - 62 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 262 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 88 T^{3} + 238 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 316 T^{4} + 64006 T^{8} - 316 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 28 T^{2} + 598 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 158 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 80 T^{2} + 2962 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 1358 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 8 T + 32 T^{2} - 88 T^{3} - 782 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 1604 T^{4} + 4130566 T^{8} + 1604 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 180 T^{2} + 14342 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 24 T + 288 T^{2} - 2376 T^{3} + 18578 T^{4} - 2376 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 13678 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 76 T^{2} + 5926 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 2404 T^{4} - 43030554 T^{8} + 2404 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 176 T^{2} + 15586 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.58399893534283187513040096404, −3.51436764306199824992737713510, −3.35742438933345171132804497824, −3.34334448309895500124209913969, −3.04466580569372588003908677121, −3.02967353794630382638203078242, −2.93314438609289184865181972499, −2.89246746512536524182660959051, −2.74193491351231495445569640391, −2.74173099937404402802733763340, −2.48028428941003334262385647635, −2.46440129233682598694503575006, −2.31352081332374865945550401254, −1.97569514414404362287309577312, −1.77689977644250628307402449092, −1.76245593882595355539864275623, −1.56980678644799949010301680013, −1.26536145643358677044102245038, −1.16413781837659209845111547137, −1.14511050173651430825594489982, −0.835392517558244729722158959324, −0.815843215051414392010794471899, −0.77273131687328518252220010391, −0.47214775554387725136258167074, −0.18708133057702897140230197810,
0.18708133057702897140230197810, 0.47214775554387725136258167074, 0.77273131687328518252220010391, 0.815843215051414392010794471899, 0.835392517558244729722158959324, 1.14511050173651430825594489982, 1.16413781837659209845111547137, 1.26536145643358677044102245038, 1.56980678644799949010301680013, 1.76245593882595355539864275623, 1.77689977644250628307402449092, 1.97569514414404362287309577312, 2.31352081332374865945550401254, 2.46440129233682598694503575006, 2.48028428941003334262385647635, 2.74173099937404402802733763340, 2.74193491351231495445569640391, 2.89246746512536524182660959051, 2.93314438609289184865181972499, 3.02967353794630382638203078242, 3.04466580569372588003908677121, 3.34334448309895500124209913969, 3.35742438933345171132804497824, 3.51436764306199824992737713510, 3.58399893534283187513040096404
Plot not available for L-functions of degree greater than 10.
