| L(s) = 1 | − 4-s + 4·9-s − 16·13-s − 3·16-s − 24·17-s + 16·29-s − 4·36-s − 18·49-s + 16·52-s + 3·64-s + 24·68-s − 24·73-s + 8·81-s + 24·97-s + 24·109-s − 16·116-s − 64·117-s + 32·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s − 96·153-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 4/3·9-s − 4.43·13-s − 3/4·16-s − 5.82·17-s + 2.97·29-s − 2/3·36-s − 2.57·49-s + 2.21·52-s + 3/8·64-s + 2.91·68-s − 2.80·73-s + 8/9·81-s + 2.43·97-s + 2.29·109-s − 1.48·116-s − 5.91·117-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s − 7.76·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08216724825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08216724825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 18 T^{2} + 162 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 16 T^{2} + 238 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 48 T^{2} + 1230 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 18 T^{2} + 1122 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 68 T^{2} + 2806 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 114 T^{2} + 6114 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 62 T^{2} + 4002 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 112 T^{2} + 6766 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 50 T^{2} + 610 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 192 T^{2} + 19230 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 296 T^{2} + 34318 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 322 T^{2} + 39682 T^{4} - 322 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.56686965044794228024914743726, −4.52133992650046334350965820473, −4.43438385690821801904350564627, −4.29559347447090011779615536570, −4.28986316344789178810004465683, −4.06145147562067464925046134502, −3.71888985637594663230214351531, −3.48789893020403113545215980199, −3.38938511181463043034400568312, −3.28319322916400185864592123567, −3.08439432726161910993295609419, −2.82047428148093689076399813832, −2.75173972481955686219411197266, −2.55407411837486769237829937892, −2.50500809853489120861107401506, −2.37973209838572399875740062732, −1.98247205072896505881950849022, −1.95850143219913165373924943942, −1.94586344830004656484747044923, −1.85970075633096684405449501223, −1.54294458480903575349032773132, −1.04775472227669861390893775886, −0.63702991383881288457738085974, −0.48984064572841265979084412540, −0.06850876168250010517407762868,
0.06850876168250010517407762868, 0.48984064572841265979084412540, 0.63702991383881288457738085974, 1.04775472227669861390893775886, 1.54294458480903575349032773132, 1.85970075633096684405449501223, 1.94586344830004656484747044923, 1.95850143219913165373924943942, 1.98247205072896505881950849022, 2.37973209838572399875740062732, 2.50500809853489120861107401506, 2.55407411837486769237829937892, 2.75173972481955686219411197266, 2.82047428148093689076399813832, 3.08439432726161910993295609419, 3.28319322916400185864592123567, 3.38938511181463043034400568312, 3.48789893020403113545215980199, 3.71888985637594663230214351531, 4.06145147562067464925046134502, 4.28986316344789178810004465683, 4.29559347447090011779615536570, 4.43438385690821801904350564627, 4.52133992650046334350965820473, 4.56686965044794228024914743726
Plot not available for L-functions of degree greater than 10.
