| L(s) = 1 | + 2-s − 3-s − 4-s − 3·5-s − 6-s − 3·7-s − 3·8-s + 9-s − 3·10-s + 12-s + 4·13-s − 3·14-s + 3·15-s − 16-s + 2·17-s + 18-s − 2·19-s + 3·20-s + 3·21-s − 7·23-s + 3·24-s + 4·25-s + 4·26-s − 27-s + 3·28-s − 8·29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.408·6-s − 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 1.10·13-s − 0.801·14-s + 0.774·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.458·19-s + 0.670·20-s + 0.654·21-s − 1.45·23-s + 0.612·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s + 0.566·28-s − 1.48·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.14714115388139726045086870267, −11.30770761943076293279755285006, −10.10747999695941287516068192337, −8.953772939637932541163524726498, −7.85280500931014021838212397954, −6.50339833314805119090611997547, −5.57200602115731485587135864806, −4.05923593712043263456563934717, −3.53695304317343710723532727862, 0,
3.53695304317343710723532727862, 4.05923593712043263456563934717, 5.57200602115731485587135864806, 6.50339833314805119090611997547, 7.85280500931014021838212397954, 8.953772939637932541163524726498, 10.10747999695941287516068192337, 11.30770761943076293279755285006, 12.14714115388139726045086870267
