| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s + 3·11-s − 12-s − 13-s + 2·14-s + 15-s + 16-s + 18-s + 7·19-s − 20-s − 2·21-s + 3·22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 2·28-s − 29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.436·21-s + 0.639·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.550714859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.550714859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.79965977526387, −13.00642007861325, −12.69852258283799, −11.92572001458676, −11.79147383664789, −11.33270999416401, −11.08165669264763, −10.22943454751541, −9.797727935266458, −9.364150272756199, −8.565580565088148, −8.061164217488890, −7.518032306847893, −7.157047837183546, −6.496710871701496, −6.005125734659399, −5.491526860235530, −4.875596176241632, −4.449019692931741, −4.040640628170748, −3.256660703134400, −2.778631101206839, −1.879390287468501, −1.219699429851781, −0.6713095798424378,
0.6713095798424378, 1.219699429851781, 1.879390287468501, 2.778631101206839, 3.256660703134400, 4.040640628170748, 4.449019692931741, 4.875596176241632, 5.491526860235530, 6.005125734659399, 6.496710871701496, 7.157047837183546, 7.518032306847893, 8.061164217488890, 8.565580565088148, 9.364150272756199, 9.797727935266458, 10.22943454751541, 11.08165669264763, 11.33270999416401, 11.79147383664789, 11.92572001458676, 12.69852258283799, 13.00642007861325, 13.79965977526387
