| L(s) = 1 | + 2·9-s − 4·19-s + 16·31-s − 48·41-s − 10·49-s + 24·59-s + 4·61-s + 20·79-s + 81-s + 48·89-s + 24·101-s + 44·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s − 8·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.917·19-s + 2.87·31-s − 7.49·41-s − 1.42·49-s + 3.12·59-s + 0.512·61-s + 2.25·79-s + 1/9·81-s + 5.08·89-s + 2.38·101-s + 4.21·109-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s − 0.611·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.644001683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644001683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 2 T - 17 T^{2} - 34 T^{3} + 4 T^{4} - 34 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4 T^{2} - 906 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 80 T^{2} + 2974 T^{4} - 79360 T^{6} - 7279805 T^{8} - 79360 p^{2} T^{10} + 2974 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 88 T^{2} + 4935 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 216 p T^{5} + 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | \( 1 - 58 T^{2} - 5807 T^{4} - 11194 T^{6} + 48855124 T^{8} - 11194 p^{2} T^{10} - 5807 p^{4} T^{12} - 58 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 98 T^{2} + 3745 T^{4} - 470302 T^{6} - 45250076 T^{8} - 470302 p^{2} T^{10} + 3745 p^{4} T^{12} + 98 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 10 T - 65 T^{2} - 70 T^{3} + 13084 T^{4} - 70 p T^{5} - 65 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 224 T^{2} + 23730 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 24 T + 272 T^{2} - 3024 T^{3} + 33231 T^{4} - 3024 p T^{5} + 272 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.78973044629388717339428297976, −3.61005810325108284499131558967, −3.47834307595929072538854844109, −3.43985344424925965053567064032, −3.38399729017221685662413001393, −3.24450951430932292387481420343, −3.23898944751189214799945577948, −3.13297758307730287525237891984, −2.86125004635927132381057750766, −2.45069437209161275042901167248, −2.44922320104061297909254333497, −2.40653219996852563387113829125, −2.32025056911417423151340820081, −2.22226998903462193636149967781, −1.93500097635828772428695326287, −1.79009295263823960500742932373, −1.78581031836153259493897039887, −1.74847173226090150153339233080, −1.32469830716755580428127488127, −1.09873134583288621434979344971, −1.02885466803301514248937818368, −0.883960705430819541517324968188, −0.794768379111883986325875775766, −0.24484853206573739426137439709, −0.17223945806252837757544083537,
0.17223945806252837757544083537, 0.24484853206573739426137439709, 0.794768379111883986325875775766, 0.883960705430819541517324968188, 1.02885466803301514248937818368, 1.09873134583288621434979344971, 1.32469830716755580428127488127, 1.74847173226090150153339233080, 1.78581031836153259493897039887, 1.79009295263823960500742932373, 1.93500097635828772428695326287, 2.22226998903462193636149967781, 2.32025056911417423151340820081, 2.40653219996852563387113829125, 2.44922320104061297909254333497, 2.45069437209161275042901167248, 2.86125004635927132381057750766, 3.13297758307730287525237891984, 3.23898944751189214799945577948, 3.24450951430932292387481420343, 3.38399729017221685662413001393, 3.43985344424925965053567064032, 3.47834307595929072538854844109, 3.61005810325108284499131558967, 3.78973044629388717339428297976
Plot not available for L-functions of degree greater than 10.
