| L(s) = 1 | − 10·3-s + 2·4-s − 2·5-s + 39·9-s − 20·12-s + 20·15-s − 4·20-s + 11·25-s − 50·27-s − 10·31-s + 78·36-s + 6·37-s − 78·45-s − 16·49-s + 8·53-s − 50·59-s + 40·60-s − 10·71-s − 110·75-s − 131·81-s + 40·89-s + 100·93-s + 34·97-s + 22·100-s + 80·103-s − 100·108-s − 60·111-s + ⋯ |
| L(s) = 1 | − 5.77·3-s + 4-s − 0.894·5-s + 13·9-s − 5.77·12-s + 5.16·15-s − 0.894·20-s + 11/5·25-s − 9.62·27-s − 1.79·31-s + 13·36-s + 0.986·37-s − 11.6·45-s − 2.28·49-s + 1.09·53-s − 6.50·59-s + 5.16·60-s − 1.18·71-s − 12.7·75-s − 14.5·81-s + 4.23·89-s + 10.3·93-s + 3.45·97-s + 11/5·100-s + 7.88·103-s − 9.62·108-s − 5.69·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06510753189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06510753189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 3 | \( ( 1 + 5 T + 2 p^{2} T^{2} + 5 p^{2} T^{3} + 89 T^{4} + 5 p^{3} T^{5} + 2 p^{4} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 9 p T^{5} - 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 16 T^{2} + 207 T^{4} + 1928 T^{6} + 15905 T^{8} + 1928 p^{2} T^{10} + 207 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 6 T^{2} + 267 T^{4} - 788 T^{6} + 50805 T^{8} - 788 p^{2} T^{10} + 267 p^{4} T^{12} - 6 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 24 T^{2} + 287 T^{4} - 48 T^{6} - 84095 T^{8} - 48 p^{2} T^{10} + 287 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 2 T^{2} + 203 T^{4} + 196 p T^{6} + 127765 T^{8} + 196 p^{3} T^{10} + 203 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 42 T^{2} + 1479 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 70 T^{2} + 3099 T^{4} - 123020 T^{6} + 4269461 T^{8} - 123020 p^{2} T^{10} + 3099 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 5 T + 56 T^{2} + 495 T^{3} + 1861 T^{4} + 495 p T^{5} + 56 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 195 p T^{5} - 28 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 90 T^{2} + 3459 T^{4} + 5660 T^{6} - 3224859 T^{8} + 5660 p^{2} T^{10} + 3459 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 72 T^{2} + 4914 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p T^{2} + 620 T^{3} - 271 T^{4} + 620 p T^{5} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + 43 T^{2} - 380 T^{3} + 4761 T^{4} - 380 p T^{5} + 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 25 T + 334 T^{2} + 3365 T^{3} + 28041 T^{4} + 3365 p T^{5} + 334 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 + 90 T^{2} + 4379 T^{4} + 59220 T^{6} - 10964459 T^{8} + 59220 p^{2} T^{10} + 4379 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 218 T^{2} + 20839 T^{4} - 218 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 5 T + 76 T^{2} - 5 T^{3} + 3981 T^{4} - 5 p T^{5} + 76 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 186 T^{2} + 8067 T^{4} - 992132 T^{6} - 141725595 T^{8} - 992132 p^{2} T^{10} + 8067 p^{4} T^{12} + 186 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 + 112 T^{2} + 17823 T^{4} + 1309064 T^{6} + 149041025 T^{8} + 1309064 p^{2} T^{10} + 17823 p^{4} T^{12} + 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 56 T^{2} - 5353 T^{4} + 476552 T^{6} + 12783505 T^{8} + 476552 p^{2} T^{10} - 5353 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 17 T + 152 T^{2} - 2215 T^{3} + 31231 T^{4} - 2215 p T^{5} + 152 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.86023777155904998113681016840, −4.77599457935239378084547797467, −4.76006461337071178498512662074, −4.66957264884010271877926749206, −4.58942125707613437714486185732, −4.34833912567528812894964646108, −4.22407245687505044085428778237, −3.84844396892410427949634597806, −3.64843020958318829531982846857, −3.37915738050577564218092891376, −3.27993092574559173088819650637, −3.20174316043294389127308529287, −3.15440923133172757892000960238, −3.00966129721995029551154919307, −2.89159042848286509660758954132, −2.40444350798597686517610760063, −2.18833498128776520191462935781, −1.89608892461994487932995836681, −1.87860284050908243641423526056, −1.87642199758636016587205063327, −1.06786450387491067407588217703, −0.871786052848504716402115339836, −0.828838749386771214316686738591, −0.34875497473216380773727359443, −0.24761847178011589772692315268,
0.24761847178011589772692315268, 0.34875497473216380773727359443, 0.828838749386771214316686738591, 0.871786052848504716402115339836, 1.06786450387491067407588217703, 1.87642199758636016587205063327, 1.87860284050908243641423526056, 1.89608892461994487932995836681, 2.18833498128776520191462935781, 2.40444350798597686517610760063, 2.89159042848286509660758954132, 3.00966129721995029551154919307, 3.15440923133172757892000960238, 3.20174316043294389127308529287, 3.27993092574559173088819650637, 3.37915738050577564218092891376, 3.64843020958318829531982846857, 3.84844396892410427949634597806, 4.22407245687505044085428778237, 4.34833912567528812894964646108, 4.58942125707613437714486185732, 4.66957264884010271877926749206, 4.76006461337071178498512662074, 4.77599457935239378084547797467, 4.86023777155904998113681016840
Plot not available for L-functions of degree greater than 10.
