| L(s) = 1 | + 40·25-s − 92·49-s − 776·73-s + 872·97-s − 248·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.24e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 8/5·25-s − 1.87·49-s − 10.6·73-s + 8.98·97-s − 2.04·121-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 + 7.36·169-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 + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.756747365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.756747365\) |
| \(L(2)\) |
|
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 | \( ( 1 - 2 p T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 + 23 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 311 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 398 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 193 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 998 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 722 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 + 1874 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1871 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3118 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 2254 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 1478 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 1858 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 3767 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 2471 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 9842 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 97 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 + 7919 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 89 | \( ( 1 + 14578 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 109 T + p^{2} T^{2} )^{8} \) |
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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
−3.68649452024599399459345860917, −3.58465916031585048581231820290, −3.34384974649480329644939636349, −3.28318414762913400511770981052, −3.18203085704647042637715155335, −3.07559832881753765004581302125, −2.99088914252751672436658939537, −2.85316628619367654128585351797, −2.68249536467048353089541738969, −2.67480374632389948999105020585, −2.65293148383394911633890595707, −2.12637740653143087012634444035, −2.02950219550437015266969637470, −1.94303370916308223012812140664, −1.88839012975244948230482070311, −1.73262354478155456360280708427, −1.57695852222697251980448136973, −1.39275221704249537170509090788, −1.24167143392576385544126125643, −1.10940976364369117149533880486, −0.793960035295608628034822420869, −0.66200582221787305803118704566, −0.54754506055407886212326282791, −0.31776103299037399007896110072, −0.11338932234674542337733753142,
0.11338932234674542337733753142, 0.31776103299037399007896110072, 0.54754506055407886212326282791, 0.66200582221787305803118704566, 0.793960035295608628034822420869, 1.10940976364369117149533880486, 1.24167143392576385544126125643, 1.39275221704249537170509090788, 1.57695852222697251980448136973, 1.73262354478155456360280708427, 1.88839012975244948230482070311, 1.94303370916308223012812140664, 2.02950219550437015266969637470, 2.12637740653143087012634444035, 2.65293148383394911633890595707, 2.67480374632389948999105020585, 2.68249536467048353089541738969, 2.85316628619367654128585351797, 2.99088914252751672436658939537, 3.07559832881753765004581302125, 3.18203085704647042637715155335, 3.28318414762913400511770981052, 3.34384974649480329644939636349, 3.58465916031585048581231820290, 3.68649452024599399459345860917
Plot not available for L-functions of degree greater than 10.
