| L(s) = 1 | − 4·4-s + 10·16-s + 8·25-s − 80·37-s − 28·49-s − 20·64-s − 96·67-s − 32·100-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 320·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 112·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 5/2·16-s + 8/5·25-s − 13.1·37-s − 4·49-s − 5/2·64-s − 11.7·67-s − 3.19·100-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 26.3·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 8·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1448722054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1448722054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| 11 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 10 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 47 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.00671049651668480183946595507, −3.94593324763705400646491978847, −3.86349426662569053080318838580, −3.73108791247670040645742355713, −3.39106934616950427853007332756, −3.31852628811714322182090010487, −3.30649132838705638537234342111, −3.29741973019958222012984350847, −3.10032691704054380442857380130, −2.99288836353459696606708235464, −2.97095095659529107701584463775, −2.78903609315232174929372207058, −2.63812848792814824318835541738, −2.11156549984205170388598857141, −1.94655416429873027856829605912, −1.89506936683780539690135376732, −1.69816078842261491857510186386, −1.59516352875141401837492155138, −1.57547708011735467768548439087, −1.46352593456786842154223815608, −1.37960647309059414548772509229, −0.935510447391173674593521508826, −0.36838899433603541896721348142, −0.18878989203895862696188751512, −0.18621788739744660289619857472,
0.18621788739744660289619857472, 0.18878989203895862696188751512, 0.36838899433603541896721348142, 0.935510447391173674593521508826, 1.37960647309059414548772509229, 1.46352593456786842154223815608, 1.57547708011735467768548439087, 1.59516352875141401837492155138, 1.69816078842261491857510186386, 1.89506936683780539690135376732, 1.94655416429873027856829605912, 2.11156549984205170388598857141, 2.63812848792814824318835541738, 2.78903609315232174929372207058, 2.97095095659529107701584463775, 2.99288836353459696606708235464, 3.10032691704054380442857380130, 3.29741973019958222012984350847, 3.30649132838705638537234342111, 3.31852628811714322182090010487, 3.39106934616950427853007332756, 3.73108791247670040645742355713, 3.86349426662569053080318838580, 3.94593324763705400646491978847, 4.00671049651668480183946595507
Plot not available for L-functions of degree greater than 10.
