| L(s) = 1 | − 5·4-s − 32·13-s + 16-s − 76·25-s − 224·37-s + 80·49-s + 160·52-s − 32·61-s + 35·64-s + 376·73-s + 928·97-s + 380·100-s − 320·109-s + 416·121-s + 127-s + 131-s + 137-s + 139-s + 1.12e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 548·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 5/4·4-s − 2.46·13-s + 1/16·16-s − 3.03·25-s − 6.05·37-s + 1.63·49-s + 3.07·52-s − 0.524·61-s + 0.546·64-s + 5.15·73-s + 9.56·97-s + 19/5·100-s − 2.93·109-s + 3.43·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7.56·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.24·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9142495360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9142495360\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5 T^{2} + 3 p^{3} T^{4} + 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 38 T^{2} + 699 T^{4} + 38 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 40 T^{2} - 498 T^{4} - 40 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 208 T^{2} + 39870 T^{4} - 208 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T + 297 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 398 T^{2} + 115443 T^{4} + 398 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 616 T^{2} + 353454 T^{4} - 616 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1360 T^{2} + 879582 T^{4} - 1360 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2942 T^{2} + 3563811 T^{4} + 2942 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1300 T^{2} + 952614 T^{4} - 1300 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 56 T + 3465 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 6548 T^{2} + 16366950 T^{4} + 6548 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 3416 T^{2} + 8731374 T^{4} + 3416 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 3460 T^{2} + 11818374 T^{4} - 3460 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 5492 T^{2} + 15934278 T^{4} + 5492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12580 T^{2} + 63740454 T^{4} - 12580 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 4665 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 12904 T^{2} + 75872814 T^{4} - 12904 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 7312 T^{2} + 62463966 T^{4} - 7312 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 94 T + 12639 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 15496 T^{2} + 116442894 T^{4} - 15496 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 26452 T^{2} + 269840070 T^{4} - 26452 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 15758 T^{2} + 176527923 T^{4} + 15758 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 232 T + 32046 T^{2} - 232 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−4.93082604839774756340402680562, −4.90373246276633966055448764855, −4.80631124216408159900753994414, −4.49190541269092667675327777006, −4.39081759992255076484578495371, −4.20306811086385230895426254230, −3.97040844880692093880716849637, −3.74679622592689173663871311934, −3.73831253445245602034984708371, −3.43558101809246062752690181166, −3.43506673677200779047436209895, −3.35742988412877312827266130595, −3.32312803835878458217979597440, −2.72780206463235163396160758525, −2.49534427979192537425559167714, −2.33707811460996534759252805091, −2.19725926329778015278040343515, −2.09407421549680295836108363215, −2.01523477691659334464871171593, −1.59155457481879692736126561089, −1.55961979610414486353451626960, −1.00601922264511626470097903453, −0.55619794445338849091873166342, −0.36748067755185164806790539332, −0.24954303309184439866212039819,
0.24954303309184439866212039819, 0.36748067755185164806790539332, 0.55619794445338849091873166342, 1.00601922264511626470097903453, 1.55961979610414486353451626960, 1.59155457481879692736126561089, 2.01523477691659334464871171593, 2.09407421549680295836108363215, 2.19725926329778015278040343515, 2.33707811460996534759252805091, 2.49534427979192537425559167714, 2.72780206463235163396160758525, 3.32312803835878458217979597440, 3.35742988412877312827266130595, 3.43506673677200779047436209895, 3.43558101809246062752690181166, 3.73831253445245602034984708371, 3.74679622592689173663871311934, 3.97040844880692093880716849637, 4.20306811086385230895426254230, 4.39081759992255076484578495371, 4.49190541269092667675327777006, 4.80631124216408159900753994414, 4.90373246276633966055448764855, 4.93082604839774756340402680562
Plot not available for L-functions of degree greater than 10.
