| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 2·13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 2·19-s + 20-s − 21-s + 4·22-s − 23-s + 24-s + 25-s + 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.218·21-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.631025834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.631025834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.86125220964856, −13.67046089524859, −13.08453973097776, −12.51474031572386, −12.11945057401879, −11.61816564322029, −11.04867732062947, −10.39609972250450, −10.02625019984951, −9.448414609193654, −8.934435219298506, −8.291334693538015, −8.119760529949156, −6.928406044150355, −6.774749936310262, −6.363421056576912, −5.728791347964053, −5.019044951597466, −4.423494078905480, −3.960604457218700, −3.236762907674622, −2.944743971849290, −1.963096423696987, −1.598707981783375, −0.7057070624066350,
0.7057070624066350, 1.598707981783375, 1.963096423696987, 2.944743971849290, 3.236762907674622, 3.960604457218700, 4.423494078905480, 5.019044951597466, 5.728791347964053, 6.363421056576912, 6.774749936310262, 6.928406044150355, 8.119760529949156, 8.291334693538015, 8.934435219298506, 9.448414609193654, 10.02625019984951, 10.39609972250450, 11.04867732062947, 11.61816564322029, 12.11945057401879, 12.51474031572386, 13.08453973097776, 13.67046089524859, 13.86125220964856
