| L(s) = 1 | + 92·25-s + 4·49-s + 200·73-s + 760·97-s − 284·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.35e3·169-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 | + 3.67·25-s + 4/49·49-s + 2.73·73-s + 7.83·97-s − 2.34·121-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 + 8·169-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(21.45703486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.45703486\) |
| \(L(2)\) |
|
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 | \( ( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | \( ( 1 - 2 T^{2} - 2397 T^{4} - 2 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 142 T^{2} + 5523 T^{4} + 142 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 17 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 19 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 23 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 29 | \( ( 1 - 50 T + p^{2} T^{2} )^{4}( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 31 | \( ( 1 + 478 T^{2} - 695037 T^{4} + 478 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 41 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 47 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 53 | \( ( 1 - 94 T + 6027 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 94 T + 6027 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | \( ( 1 - 6862 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 67 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 71 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 73 | \( ( 1 - 50 T - 2829 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 9118 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 4178 T^{2} - 30002637 T^{4} - 4178 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 97 | \( ( 1 - 190 T + 26691 T^{2} - 190 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−3.79689671805417718512898171467, −3.35344150433852542322232328172, −3.35164476401223321120602136866, −3.33908590069716866549716745599, −3.27409818666700371604734190803, −3.26419613613599966692003804427, −2.96983775772443191495723507365, −2.75684779485349219653072775597, −2.74926773589679613739063475190, −2.61622407093589162073689523693, −2.39550814104313377710871444034, −2.29988315488131069726556744327, −2.04010529517000824752362310769, −2.02746237702728661581668228540, −1.89968797854150380301015951127, −1.83600833824568443042964103510, −1.48456698264684509174093350152, −1.36154866520004814764230080448, −1.11469235604203576448167252777, −0.987190930114747468350951789690, −0.922920546563537141627362439046, −0.63608244891083571944043796807, −0.61806009169192532007274278592, −0.35683936605372028514517666986, −0.27469055374492088567566089264,
0.27469055374492088567566089264, 0.35683936605372028514517666986, 0.61806009169192532007274278592, 0.63608244891083571944043796807, 0.922920546563537141627362439046, 0.987190930114747468350951789690, 1.11469235604203576448167252777, 1.36154866520004814764230080448, 1.48456698264684509174093350152, 1.83600833824568443042964103510, 1.89968797854150380301015951127, 2.02746237702728661581668228540, 2.04010529517000824752362310769, 2.29988315488131069726556744327, 2.39550814104313377710871444034, 2.61622407093589162073689523693, 2.74926773589679613739063475190, 2.75684779485349219653072775597, 2.96983775772443191495723507365, 3.26419613613599966692003804427, 3.27409818666700371604734190803, 3.33908590069716866549716745599, 3.35164476401223321120602136866, 3.35344150433852542322232328172, 3.79689671805417718512898171467
Plot not available for L-functions of degree greater than 10.
