| L(s) = 1 | + 12·11-s + 2·16-s − 48·31-s − 108·61-s − 96·71-s − 7·81-s − 36·101-s + 94·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 24·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 1/2·16-s − 8.62·31-s − 13.8·61-s − 11.3·71-s − 7/9·81-s − 3.58·101-s + 8.54·121-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 + 0.0760·173-s + 1.80·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09256163904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09256163904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 73 T^{4} + 2928 T^{8} + 73 p^{4} T^{12} + p^{8} T^{16} \) |
| good | 3 | \( 1 + 7 T^{4} - 37 p T^{8} - 14 T^{12} + 15070 T^{16} - 14 p^{4} T^{20} - 37 p^{9} T^{24} + 7 p^{12} T^{28} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + T^{4} - 50736 T^{8} + p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 94 T^{4} - 74685 T^{8} + 94 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 11 T^{2} - 503 T^{4} - 1078 T^{6} + 215374 T^{8} - 1078 p^{2} T^{10} - 503 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 311 T^{4} - 183120 T^{8} - 311 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 83 T^{2} + 3276 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{8} \) |
| 37 | \( 1 - 1393 T^{4} - 596879 T^{8} + 1686914642 T^{12} - 711159912626 T^{16} + 1686914642 p^{4} T^{20} - 596879 p^{8} T^{24} - 1393 p^{12} T^{28} + p^{16} T^{32} \) |
| 41 | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{8} \) |
| 43 | \( ( 1 + 217 T^{4} - 5396064 T^{8} + 217 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 47 | \( 1 - 4153 T^{4} + 8461249 T^{8} + 4041707906 T^{12} - 30776533112690 T^{16} + 4041707906 p^{4} T^{20} + 8461249 p^{8} T^{24} - 4153 p^{12} T^{28} + p^{16} T^{32} \) |
| 53 | \( 1 + 7727 T^{4} + 31770289 T^{8} + 93923833106 T^{12} + 247060582275790 T^{16} + 93923833106 p^{4} T^{20} + 31770289 p^{8} T^{24} + 7727 p^{12} T^{28} + p^{16} T^{32} \) |
| 59 | \( ( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 + 27 T + 383 T^{2} + 3780 T^{3} + 30702 T^{4} + 3780 p T^{5} + 383 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 67 | \( 1 + 3479 T^{4} + 20488945 T^{8} - 169384668334 T^{12} - 579648904329986 T^{16} - 169384668334 p^{4} T^{20} + 20488945 p^{8} T^{24} + 3479 p^{12} T^{28} + p^{16} T^{32} \) |
| 71 | \( ( 1 + 6 T + p T^{2} )^{16} \) |
| 73 | \( 1 - 2788 T^{4} + 31113226 T^{8} + 223421298032 T^{12} - 482429265046061 T^{16} + 223421298032 p^{4} T^{20} + 31113226 p^{8} T^{24} - 2788 p^{12} T^{28} + p^{16} T^{32} \) |
| 79 | \( ( 1 + 58 T^{2} - 2877 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 1033 T^{4} - 67132992 T^{8} + 1033 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 257 T^{2} + 34849 T^{4} - 3947006 T^{6} + 387523630 T^{8} - 3947006 p^{2} T^{10} + 34849 p^{4} T^{12} - 257 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 28228 T^{4} + 363130758 T^{8} + 28228 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
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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
−3.14883145304024806619796707490, −3.08452941630768717416033346834, −3.05192753025677375296128214643, −2.98293181514568086358181187511, −2.97197847290170583467893458928, −2.83669054929495942983813908196, −2.80475733430882850047103596345, −2.73861315938511318242733077391, −2.61915666940489464675295935097, −2.44614033664507478815000815880, −2.07903228808015479276938913898, −1.92243054662513172381906332488, −1.81949803846143164268510012797, −1.80933723055707531416643263303, −1.77004182625294183000439212596, −1.73971307351220023775841764390, −1.57251975528730482051901351690, −1.57239003397616868963923840520, −1.53034941847956183385969526463, −1.39117435231596437737297430294, −1.28739932371753801130771945720, −1.11389925869893923053752047402, −0.60155471473086333610227946179, −0.27137996911791011161796677259, −0.06192247636326251883632669373,
0.06192247636326251883632669373, 0.27137996911791011161796677259, 0.60155471473086333610227946179, 1.11389925869893923053752047402, 1.28739932371753801130771945720, 1.39117435231596437737297430294, 1.53034941847956183385969526463, 1.57239003397616868963923840520, 1.57251975528730482051901351690, 1.73971307351220023775841764390, 1.77004182625294183000439212596, 1.80933723055707531416643263303, 1.81949803846143164268510012797, 1.92243054662513172381906332488, 2.07903228808015479276938913898, 2.44614033664507478815000815880, 2.61915666940489464675295935097, 2.73861315938511318242733077391, 2.80475733430882850047103596345, 2.83669054929495942983813908196, 2.97197847290170583467893458928, 2.98293181514568086358181187511, 3.05192753025677375296128214643, 3.08452941630768717416033346834, 3.14883145304024806619796707490
Plot not available for L-functions of degree greater than 10.
