| L(s) = 1 | + 4·25-s + 16·49-s + 16·61-s + 16·109-s − 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4/5·25-s + 16/7·49-s + 2.04·61-s + 1.53·109-s − 5.81·121-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 + 6.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.799908713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799908713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.74085664139760024224437962530, −3.50898221169824766182126800332, −3.30896075098848865546844955685, −3.11829462171938108531380550501, −3.06226194867756057401765268584, −3.04327326930013401793778065992, −3.04172708649602244484488712535, −2.99339865455116542788334488847, −2.64579753261378573513404106629, −2.58022570843821904992918452196, −2.46050063143127897782753690368, −2.15345968698440373248261369059, −2.07569507470788890516003616365, −2.05811285531847110294435860478, −1.96096036087451722990762698319, −1.93658894330022312931645099829, −1.64718594128255049594050552311, −1.26741593950734141883962317805, −1.23729459403020196778518069708, −1.23250837053924739108077370764, −0.842608980815016461705961192752, −0.790762929631313492287406140967, −0.75861915184940834965947655636, −0.34690760557934445915663844958, −0.13822761797608785625732089284,
0.13822761797608785625732089284, 0.34690760557934445915663844958, 0.75861915184940834965947655636, 0.790762929631313492287406140967, 0.842608980815016461705961192752, 1.23250837053924739108077370764, 1.23729459403020196778518069708, 1.26741593950734141883962317805, 1.64718594128255049594050552311, 1.93658894330022312931645099829, 1.96096036087451722990762698319, 2.05811285531847110294435860478, 2.07569507470788890516003616365, 2.15345968698440373248261369059, 2.46050063143127897782753690368, 2.58022570843821904992918452196, 2.64579753261378573513404106629, 2.99339865455116542788334488847, 3.04172708649602244484488712535, 3.04327326930013401793778065992, 3.06226194867756057401765268584, 3.11829462171938108531380550501, 3.30896075098848865546844955685, 3.50898221169824766182126800332, 3.74085664139760024224437962530
Plot not available for L-functions of degree greater than 10.
