| L(s) = 1 | + 16·31-s + 112·61-s + 9·81-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 2.87·31-s + 14.3·61-s + 81-s − 1.81·121-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 + 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 + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.65368852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.65368852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 958 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 2471 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 5617 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 10871 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 2498 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.03717477391817073978773829894, −3.96309375810151982486713712498, −3.95645186162068615599197660337, −3.70053808123275081047376341798, −3.68951848451044933784959659385, −3.58560625274578775274884648650, −3.50567765818631183441572520871, −3.27804777645965682900312012468, −3.16957649012807775765798560959, −2.80317407949117363059235309587, −2.73185524803451681764178598294, −2.63002179543864579666181413711, −2.48225459335107408831017657824, −2.44983527424237407501867491656, −2.27709391163683973988392863191, −2.18472261733681280352977901857, −1.91130208174629345261707683904, −1.77815878357079756629660900659, −1.70364361150028875035233890398, −1.25569933152877174065528717201, −1.01964235625341735660832326226, −0.858150899837855139963749793619, −0.73364833704205960273157373076, −0.73228558752995074721702968203, −0.37038368828786188693121210043,
0.37038368828786188693121210043, 0.73228558752995074721702968203, 0.73364833704205960273157373076, 0.858150899837855139963749793619, 1.01964235625341735660832326226, 1.25569933152877174065528717201, 1.70364361150028875035233890398, 1.77815878357079756629660900659, 1.91130208174629345261707683904, 2.18472261733681280352977901857, 2.27709391163683973988392863191, 2.44983527424237407501867491656, 2.48225459335107408831017657824, 2.63002179543864579666181413711, 2.73185524803451681764178598294, 2.80317407949117363059235309587, 3.16957649012807775765798560959, 3.27804777645965682900312012468, 3.50567765818631183441572520871, 3.58560625274578775274884648650, 3.68951848451044933784959659385, 3.70053808123275081047376341798, 3.95645186162068615599197660337, 3.96309375810151982486713712498, 4.03717477391817073978773829894
Plot not available for L-functions of degree greater than 10.
