| L(s) = 1 | − 80·25-s + 128·31-s − 184·49-s − 160·73-s + 384·79-s − 192·97-s − 896·103-s − 360·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 408·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.19·25-s + 4.12·31-s − 3.75·49-s − 2.19·73-s + 4.86·79-s − 1.97·97-s − 8.69·103-s − 2.97·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.41·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.691748700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.691748700\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 8 p T^{2} + 818 T^{4} + 8 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 180 T^{2} + 24070 T^{4} + 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 204 T^{2} + 14278 T^{4} - 204 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 320 T^{2} + 189314 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 60 T^{2} + 48550 T^{4} + 60 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1652 T^{2} + 1188710 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2376 T^{2} + 2685298 T^{4} + 2376 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 32 T + 1710 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 3972 T^{2} + 7479526 T^{4} - 3972 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 5792 T^{2} + 13955138 T^{4} - 5792 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 996 T^{2} + 6872614 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4660 T^{2} + 10875174 T^{4} - 4660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8712 T^{2} + 34754866 T^{4} + 8712 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 9316 T^{2} + 45079718 T^{4} + 9316 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 9412 T^{2} + 45525030 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 12484 T^{2} + 74951718 T^{4} - 12484 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 9076 T^{2} + 54164454 T^{4} - 9076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 40 T + 7730 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 96 T + 8494 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 24436 T^{2} + 241946438 T^{4} + 24436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 25856 T^{2} + 284297666 T^{4} - 25856 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 48 T + 18562 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−4.93830716689617328820092122830, −4.93171713760566414678758864516, −4.64538288003719821493114741381, −4.63776127653140892912084347502, −4.33775325318356555216943537377, −4.30982402587458502723437005672, −4.29202985879768676313889961733, −3.89636018129713130459275301304, −3.64236037376070479338388046642, −3.61016671107641941116586807777, −3.54842404351342397979775047733, −3.51624532274301996575875238235, −2.83717381964411544984120605253, −2.81134679450415079759435729922, −2.68171641774824260029629214425, −2.68114372964456075806356629471, −2.49525561757794854702562045339, −2.16769381196155877109088458642, −1.66259606934885322690390387991, −1.50736534809050241040141801392, −1.50658049238568277020740302238, −1.50250572812226948328656156600, −0.65728018669777348261118628501, −0.62383270314453365097414537850, −0.18111100150737372037929251956,
0.18111100150737372037929251956, 0.62383270314453365097414537850, 0.65728018669777348261118628501, 1.50250572812226948328656156600, 1.50658049238568277020740302238, 1.50736534809050241040141801392, 1.66259606934885322690390387991, 2.16769381196155877109088458642, 2.49525561757794854702562045339, 2.68114372964456075806356629471, 2.68171641774824260029629214425, 2.81134679450415079759435729922, 2.83717381964411544984120605253, 3.51624532274301996575875238235, 3.54842404351342397979775047733, 3.61016671107641941116586807777, 3.64236037376070479338388046642, 3.89636018129713130459275301304, 4.29202985879768676313889961733, 4.30982402587458502723437005672, 4.33775325318356555216943537377, 4.63776127653140892912084347502, 4.64538288003719821493114741381, 4.93171713760566414678758864516, 4.93830716689617328820092122830
Plot not available for L-functions of degree greater than 10.
