| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 2·7-s − 3·8-s + 9-s − 12-s − 2·13-s − 2·14-s − 16-s + 18-s + 4·19-s − 2·21-s + 23-s − 3·24-s − 2·26-s + 27-s + 2·28-s − 4·29-s + 5·32-s − 36-s − 2·37-s + 4·38-s − 2·39-s − 4·41-s − 2·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.208·23-s − 0.612·24-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.742·29-s + 0.883·32-s − 1/6·36-s − 0.328·37-s + 0.648·38-s − 0.320·39-s − 0.624·41-s − 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.28340808635157, −12.93540059287706, −12.41715591377661, −12.08151884427378, −11.49862268818983, −11.07813176357095, −10.12912846308737, −9.903045721427726, −9.631916978567122, −8.969695952954687, −8.580327782059112, −8.181416092087840, −7.397778809128558, −7.009995045785625, −6.610923068066844, −5.782631645342593, −5.469196138923764, −4.993924579477822, −4.358694690911181, −3.767721317287763, −3.483444743590081, −2.845364559897522, −2.437293741928117, −1.582216133737111, −0.7435283500706543, 0,
0.7435283500706543, 1.582216133737111, 2.437293741928117, 2.845364559897522, 3.483444743590081, 3.767721317287763, 4.358694690911181, 4.993924579477822, 5.469196138923764, 5.782631645342593, 6.610923068066844, 7.009995045785625, 7.397778809128558, 8.181416092087840, 8.580327782059112, 8.969695952954687, 9.631916978567122, 9.903045721427726, 10.12912846308737, 11.07813176357095, 11.49862268818983, 12.08151884427378, 12.41715591377661, 12.93540059287706, 13.28340808635157
