| L(s) = 1 | − 8·4-s + 36·16-s − 16·25-s − 16·37-s − 80·43-s + 12·49-s − 120·64-s + 16·67-s + 96·79-s + 128·100-s + 112·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 640·172-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4·4-s + 9·16-s − 3.19·25-s − 2.63·37-s − 12.1·43-s + 12/7·49-s − 15·64-s + 1.95·67-s + 10.8·79-s + 64/5·100-s + 10.7·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 48.7·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1003374081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1003374081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 6 T^{2} - 8 T^{3} + 66 T^{4} - 8 p T^{5} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + T^{2} )^{8} \) |
| good | 5 | \( ( 1 + 8 T^{2} + 66 T^{4} + 88 p T^{6} + 1922 T^{8} + 88 p^{3} T^{10} + 66 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 444 T^{4} - 6256 T^{6} + 99494 T^{8} - 6256 p^{2} T^{10} + 444 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 84 T^{2} + 3462 T^{4} + 94052 T^{6} + 1855362 T^{8} + 94052 p^{2} T^{10} + 3462 p^{4} T^{12} + 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 96 T^{2} + 3954 T^{4} - 98080 T^{6} + 1928610 T^{8} - 98080 p^{2} T^{10} + 3954 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - p T^{2} )^{16} \) |
| 29 | \( ( 1 - 44 T^{2} + 1256 T^{4} - 35556 T^{6} + 935342 T^{8} - 35556 p^{2} T^{10} + 1256 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 156 T^{2} + 12630 T^{4} - 662044 T^{6} + 24293346 T^{8} - 662044 p^{2} T^{10} + 12630 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 4 T + 72 T^{2} + 220 T^{3} + 3038 T^{4} + 220 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 212 T^{2} + 20550 T^{4} + 1257668 T^{6} + 57610946 T^{8} + 1257668 p^{2} T^{10} + 20550 p^{4} T^{12} + 212 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 + 20 T + 182 T^{2} + 900 T^{3} + 4162 T^{4} + 900 p T^{5} + 182 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 236 T^{2} + 27318 T^{4} + 2044556 T^{6} + 110968226 T^{8} + 2044556 p^{2} T^{10} + 27318 p^{4} T^{12} + 236 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 140 T^{2} + 14696 T^{4} - 1012164 T^{6} + 62083310 T^{8} - 1012164 p^{2} T^{10} + 14696 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 56 T^{2} + 7644 T^{4} + 5848 p T^{6} + 26980262 T^{8} + 5848 p^{3} T^{10} + 7644 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 288 T^{2} + 36636 T^{4} - 2886112 T^{6} + 182583654 T^{8} - 2886112 p^{2} T^{10} + 36636 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 4 T + 182 T^{2} - 948 T^{3} + 15554 T^{4} - 948 p T^{5} + 182 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{8} \) |
| 73 | \( ( 1 - 100 T^{2} + 24486 T^{4} - 1625908 T^{6} + 204031490 T^{8} - 1625908 p^{2} T^{10} + 24486 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 24 T + 384 T^{2} - 4648 T^{3} + 46626 T^{4} - 4648 p T^{5} + 384 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 188 T^{2} + 11094 T^{4} - 614020 T^{6} - 134700574 T^{8} - 614020 p^{2} T^{10} + 11094 p^{4} T^{12} + 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 212 T^{2} + 19656 T^{4} + 576572 T^{6} - 23116594 T^{8} + 576572 p^{2} T^{10} + 19656 p^{4} T^{12} + 212 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 520 T^{2} + 135324 T^{4} - 22521400 T^{6} + 2595040838 T^{8} - 22521400 p^{2} T^{10} + 135324 p^{4} T^{12} - 520 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.26399579118988848309903113561, −2.21200436451613776363622875608, −2.19991909662191238484008084676, −2.19643120191551305692898175386, −2.18850716716216144721788529402, −2.00424207055247319798801922066, −1.96462375109688616397068720914, −1.93657718648323871182440296832, −1.79039510803660460522198372216, −1.75571355858812429980630314109, −1.67462866062675116638573537678, −1.59932473615433069307877982831, −1.44433532799006207901965941232, −1.40705272634629840566816131325, −1.23478675821366208380600099169, −1.06127850023370383166992168535, −0.994323583622214011312277991141, −0.896942950603761679239628991824, −0.896588221863029413990816618594, −0.797478043415098900186180887723, −0.48575341743505741259462054025, −0.47374750025638383676536655349, −0.20463531435619907560049729529, −0.18773704820667988327233071975, −0.087808119439193485854068825502,
0.087808119439193485854068825502, 0.18773704820667988327233071975, 0.20463531435619907560049729529, 0.47374750025638383676536655349, 0.48575341743505741259462054025, 0.797478043415098900186180887723, 0.896588221863029413990816618594, 0.896942950603761679239628991824, 0.994323583622214011312277991141, 1.06127850023370383166992168535, 1.23478675821366208380600099169, 1.40705272634629840566816131325, 1.44433532799006207901965941232, 1.59932473615433069307877982831, 1.67462866062675116638573537678, 1.75571355858812429980630314109, 1.79039510803660460522198372216, 1.93657718648323871182440296832, 1.96462375109688616397068720914, 2.00424207055247319798801922066, 2.18850716716216144721788529402, 2.19643120191551305692898175386, 2.19991909662191238484008084676, 2.21200436451613776363622875608, 2.26399579118988848309903113561
Plot not available for L-functions of degree greater than 10.
