| L(s) = 1 | − 3·4-s − 4·5-s − 10·9-s + 3·16-s − 8·17-s + 12·20-s − 14·25-s + 30·36-s + 40·45-s + 20·49-s + 44·61-s − 3·64-s + 24·68-s + 32·73-s − 12·80-s + 35·81-s + 32·85-s + 42·100-s + 58·121-s + 80·125-s + 127-s + 131-s + 137-s + 139-s − 30·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.78·5-s − 3.33·9-s + 3/4·16-s − 1.94·17-s + 2.68·20-s − 2.79·25-s + 5·36-s + 5.96·45-s + 20/7·49-s + 5.63·61-s − 3/8·64-s + 2.91·68-s + 3.74·73-s − 1.34·80-s + 35/9·81-s + 3.47·85-s + 21/5·100-s + 5.27·121-s + 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/2·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08968322008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08968322008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3 T^{2} + 3 p T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | \( 1 - 16 T^{2} + 718 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 5 T^{2} + 20 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 10 T^{2} + 55 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 29 T^{2} + 448 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 11 T^{2} + 28 p T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + T + p T^{2} )^{8} \) |
| 23 | \( ( 1 - 73 T^{2} + 2352 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 43 T^{2} + 916 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 104 T^{2} + 4558 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 104 T^{2} + 5374 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 120 T^{2} + 6894 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 49 T^{2} + 724 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 53 T^{2} + 4776 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 115 T^{2} + 6676 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 61 T^{2} + 5644 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 11 T + 114 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 173 T^{2} + 14212 T^{4} + 173 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 160 T^{2} + 13150 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 145 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 72 T^{2} + 10446 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 96 T^{2} + 15470 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 124 T^{2} + 12886 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 96 T^{2} + 1470 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−7.01961326528275369202131514975, −6.76656881913769197688988453338, −6.58246677224504565295475307962, −6.48062614511112637960427365530, −6.24193810891898516110196596595, −5.88743290187322996826442695187, −5.73687651343927465313273336896, −5.71508598516158499356213175893, −5.49894328770866825014555193549, −5.42207377499383622179197897989, −5.17037024768563289616503596796, −4.91263505069066043134884676055, −4.50781930093701196262537216478, −4.42148515038659471959616264176, −4.42090739359444318965534173660, −3.84277838462172854714911291787, −3.83439248286397627582337704208, −3.64174237733015602439561412964, −3.62028145778217057486891019273, −3.32472764545595268802635285011, −2.67371941033333324457382277444, −2.42157600005783973186156524864, −2.27673060046378987766924326058, −2.18579312118677360840564245362, −0.43906909041208697170077648799,
0.43906909041208697170077648799, 2.18579312118677360840564245362, 2.27673060046378987766924326058, 2.42157600005783973186156524864, 2.67371941033333324457382277444, 3.32472764545595268802635285011, 3.62028145778217057486891019273, 3.64174237733015602439561412964, 3.83439248286397627582337704208, 3.84277838462172854714911291787, 4.42090739359444318965534173660, 4.42148515038659471959616264176, 4.50781930093701196262537216478, 4.91263505069066043134884676055, 5.17037024768563289616503596796, 5.42207377499383622179197897989, 5.49894328770866825014555193549, 5.71508598516158499356213175893, 5.73687651343927465313273336896, 5.88743290187322996826442695187, 6.24193810891898516110196596595, 6.48062614511112637960427365530, 6.58246677224504565295475307962, 6.76656881913769197688988453338, 7.01961326528275369202131514975
Plot not available for L-functions of degree greater than 10.
