| 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 | + 16/5·25-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.115750677\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115750677\times10^{-5}\) |
| \(L(2)\) |
|
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
−3.59888085864097632307226130175, −3.52364515146807392209260988985, −3.33620616025155586720763136514, −3.33172181967903098283051249924, −3.04216618880880090458603036595, −2.86355890642751310533398444241, −2.72941827871071753548409476662, −2.69694242440984861445954629405, −2.55049100114061369055955430629, −2.50828525692627220961548292370, −2.42803518349022408245705235083, −2.34043804638585382130870535047, −1.89712766815863935676855724508, −1.88108547611341321762756175076, −1.67569645655675776812091963598, −1.39853560623343096381039132576, −1.39829873711421742231565580609, −1.37860624715850079558914968625, −1.29487573534759250649344706111, −1.17392278174882214030952800243, −0.919291014686476275688669939781, −0.60996996095935543636524780147, −0.31108377773153312010047639246, −0.03696628152694583932163027123, −0.00297910643702144398901182126,
0.00297910643702144398901182126, 0.03696628152694583932163027123, 0.31108377773153312010047639246, 0.60996996095935543636524780147, 0.919291014686476275688669939781, 1.17392278174882214030952800243, 1.29487573534759250649344706111, 1.37860624715850079558914968625, 1.39829873711421742231565580609, 1.39853560623343096381039132576, 1.67569645655675776812091963598, 1.88108547611341321762756175076, 1.89712766815863935676855724508, 2.34043804638585382130870535047, 2.42803518349022408245705235083, 2.50828525692627220961548292370, 2.55049100114061369055955430629, 2.69694242440984861445954629405, 2.72941827871071753548409476662, 2.86355890642751310533398444241, 3.04216618880880090458603036595, 3.33172181967903098283051249924, 3.33620616025155586720763136514, 3.52364515146807392209260988985, 3.59888085864097632307226130175
Plot not available for L-functions of degree greater than 10.
