| L(s) = 1 | + 2·4-s − 12·5-s + 2·9-s + 12·11-s + 16-s + 4·19-s − 24·20-s + 70·25-s + 12·29-s − 32·31-s + 4·36-s + 16·41-s + 24·44-s − 24·45-s − 4·49-s − 144·55-s + 8·59-s − 16·61-s − 2·64-s − 20·71-s + 8·76-s + 32·79-s − 12·80-s + 81-s + 8·89-s − 48·95-s + 24·99-s + ⋯ |
| L(s) = 1 | + 4-s − 5.36·5-s + 2/3·9-s + 3.61·11-s + 1/4·16-s + 0.917·19-s − 5.36·20-s + 14·25-s + 2.22·29-s − 5.74·31-s + 2/3·36-s + 2.49·41-s + 3.61·44-s − 3.57·45-s − 4/7·49-s − 19.4·55-s + 1.04·59-s − 2.04·61-s − 1/4·64-s − 2.37·71-s + 0.917·76-s + 3.60·79-s − 1.34·80-s + 1/9·81-s + 0.847·89-s − 4.92·95-s + 2.41·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485367143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485367143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( 1 + 4 T^{2} - 6 p T^{4} - 160 T^{6} + 179 T^{8} - 160 p^{2} T^{10} - 6 p^{5} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 6 T + 16 T^{2} + 12 T^{3} - 117 T^{4} + 12 p T^{5} + 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 22 T^{2} - 39 T^{4} + 1210 T^{6} + 50132 T^{8} + 1210 p^{2} T^{10} - 39 p^{4} T^{12} - 22 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 2 T - 24 T^{2} + 20 T^{3} + 347 T^{4} + 20 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 20 T^{2} + 342 T^{4} - 20000 T^{6} - 488077 T^{8} - 20000 p^{2} T^{10} + 342 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 37 | \( 1 + 58 T^{2} - 39 T^{4} + 38570 T^{6} + 5100932 T^{8} + 38570 p^{2} T^{10} - 39 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 8 T - 23 T^{2} - 40 T^{3} + 3264 T^{4} - 40 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 148 T^{2} + 12774 T^{4} + 803936 T^{6} + 39437603 T^{8} + 803936 p^{2} T^{10} + 12774 p^{4} T^{12} + 148 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 116 T^{2} + 6682 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 62 T^{2} + 71 p T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 4 T - 62 T^{2} + 160 T^{3} + 1659 T^{4} + 160 p T^{5} - 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T - 63 T^{2} + 40 T^{3} + 8504 T^{4} + 40 p T^{5} - 63 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 52 T^{2} - 3386 T^{4} - 150176 T^{6} + 4798723 T^{8} - 150176 p^{2} T^{10} - 3386 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 10 T - 56 T^{2} + 140 T^{3} + 12195 T^{4} + 140 p T^{5} - 56 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 238 T^{2} + 24115 T^{4} - 238 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 260 T^{2} + 29578 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 4 T + 10 T^{2} + 688 T^{3} - 9309 T^{4} + 688 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.99847806096568552797168921568, −4.84028236311073486897153651876, −4.58085776287247188751385081668, −4.50644491645774969836414286676, −4.38525972764236098810736960851, −4.18691900298102542978135652686, −3.90478576317493358920774035989, −3.85054685447367894905492840334, −3.80111697215764154688015087510, −3.78592254697986734201503904384, −3.70433659246340258197483418477, −3.47051322623142593791415808095, −3.46121709858357593489477446716, −3.04655496842989850086482612621, −2.97034219962762019841902981105, −2.88444488410576779205934733830, −2.65059319305615464322597615150, −2.03887873891635708420536567741, −1.95617626393175990760011811197, −1.91566676168322572870801759898, −1.60333887787027055187157742871, −1.20596899374090844767637343385, −1.00516819281998937664595215136, −0.63359442193151651574333785910, −0.40751719853749805095678692506,
0.40751719853749805095678692506, 0.63359442193151651574333785910, 1.00516819281998937664595215136, 1.20596899374090844767637343385, 1.60333887787027055187157742871, 1.91566676168322572870801759898, 1.95617626393175990760011811197, 2.03887873891635708420536567741, 2.65059319305615464322597615150, 2.88444488410576779205934733830, 2.97034219962762019841902981105, 3.04655496842989850086482612621, 3.46121709858357593489477446716, 3.47051322623142593791415808095, 3.70433659246340258197483418477, 3.78592254697986734201503904384, 3.80111697215764154688015087510, 3.85054685447367894905492840334, 3.90478576317493358920774035989, 4.18691900298102542978135652686, 4.38525972764236098810736960851, 4.50644491645774969836414286676, 4.58085776287247188751385081668, 4.84028236311073486897153651876, 4.99847806096568552797168921568
Plot not available for L-functions of degree greater than 10.
