| L(s) = 1 | + 8·4-s − 4·9-s + 24·16-s − 8·19-s + 10·25-s − 28·31-s − 32·36-s − 28·49-s − 64·76-s + 36·79-s + 9·81-s + 80·100-s + 88·109-s + 14·121-s − 224·124-s + ⋯ |
| L(s) = 1 | + 4·4-s − 4/3·9-s + 6·16-s − 1.83·19-s + 2·25-s − 5.02·31-s − 5.33·36-s − 4·49-s − 7.34·76-s + 4.05·79-s + 81-s + 8·100-s + 8.42·109-s + 1.27·121-s − 20.1·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.248564068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.248564068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4 T^{2} + 7 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| good | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 7 T^{2} - 72 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 87 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 18 T + 145 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}( 1 + 18 T + 145 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 - 43 T^{2} + 168 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 76 T^{2} + 3927 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 74 T^{2} + 2667 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 113 T^{2} + 9288 T^{4} + 113 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 137 T^{2} + 13728 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 136 T^{2} + 13167 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 13 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 + 68 T^{2} - 2265 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.84589522911762216454749006739, −4.82626293346223415432874156828, −4.68692219400316137266462972867, −4.40840464898628282996046310417, −4.26679079884290597408035110824, −4.24154700669189483488181694634, −4.01548868161882063667816586770, −3.51437291065586706585470553232, −3.48947982135789709242328711602, −3.46846991818869695152399235358, −3.35117558196847519661444658065, −3.18757997309896305929898091842, −3.00213491012546462159679911361, −2.94149350366289492725802636819, −2.69863526886279417386427284531, −2.29546981152953530500841953949, −2.26654821321878134095280655221, −2.12882030667796330307873570479, −2.08608280139705332378166087237, −1.91249469096052626984408756621, −1.80246877291148513791654734244, −1.48419702645762294723753797447, −1.38289762789051552626628822453, −0.58953470594345482851835586679, −0.37144971994376025440294748576,
0.37144971994376025440294748576, 0.58953470594345482851835586679, 1.38289762789051552626628822453, 1.48419702645762294723753797447, 1.80246877291148513791654734244, 1.91249469096052626984408756621, 2.08608280139705332378166087237, 2.12882030667796330307873570479, 2.26654821321878134095280655221, 2.29546981152953530500841953949, 2.69863526886279417386427284531, 2.94149350366289492725802636819, 3.00213491012546462159679911361, 3.18757997309896305929898091842, 3.35117558196847519661444658065, 3.46846991818869695152399235358, 3.48947982135789709242328711602, 3.51437291065586706585470553232, 4.01548868161882063667816586770, 4.24154700669189483488181694634, 4.26679079884290597408035110824, 4.40840464898628282996046310417, 4.68692219400316137266462972867, 4.82626293346223415432874156828, 4.84589522911762216454749006739
Plot not available for L-functions of degree greater than 10.
