| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s + 4·11-s + 12-s − 6·13-s − 2·14-s + 16-s − 6·17-s + 18-s − 2·19-s − 2·21-s + 4·22-s − 4·23-s + 24-s − 6·26-s + 27-s − 2·28-s + 8·29-s + 32-s + 4·33-s − 6·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.436·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.377·28-s + 1.48·29-s + 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131530101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131530101\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.12488164591805, −12.60115717603378, −12.16771532444380, −11.69755260899705, −11.49983364220111, −10.55254373379223, −10.17361735275669, −9.837096715287038, −9.240690725141011, −8.865309215898354, −8.232174174197975, −7.856993111426086, −6.902164879799988, −6.780727651152016, −6.563870202341898, −5.782902504425397, −5.111172462852346, −4.523234162030302, −4.250679329805656, −3.661814865284420, −2.999459268316674, −2.581191405151636, −2.000929360864695, −1.439041721461618, −0.3204932389474585,
0.3204932389474585, 1.439041721461618, 2.000929360864695, 2.581191405151636, 2.999459268316674, 3.661814865284420, 4.250679329805656, 4.523234162030302, 5.111172462852346, 5.782902504425397, 6.563870202341898, 6.780727651152016, 6.902164879799988, 7.856993111426086, 8.232174174197975, 8.865309215898354, 9.240690725141011, 9.837096715287038, 10.17361735275669, 10.55254373379223, 11.49983364220111, 11.69755260899705, 12.16771532444380, 12.60115717603378, 13.12488164591805
