| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 2·7-s + 8-s + 9-s − 4·10-s + 12-s + 13-s + 2·14-s − 4·15-s + 16-s + 17-s + 18-s + 4·19-s − 4·20-s + 2·21-s − 6·23-s + 24-s + 11·25-s + 26-s + 27-s + 2·28-s − 4·30-s + 10·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.436·21-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 0.377·28-s − 0.730·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.778858298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.778858298\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.24190212836186, −12.78549762842972, −12.27485780485367, −11.82683643347978, −11.41959534873894, −11.27664537686374, −10.52578680318593, −9.982158353597886, −9.485051078285522, −8.706477186735284, −8.142326514015190, −8.022819531163040, −7.562358016553593, −7.099634979558196, −6.399228511842473, −5.921236156021143, −5.055539243280185, −4.731368453025468, −4.106772661366034, −3.835590848428691, −3.287536145676056, −2.622015086395693, −2.131530371896412, −1.020527383614924, −0.7363854429394138,
0.7363854429394138, 1.020527383614924, 2.131530371896412, 2.622015086395693, 3.287536145676056, 3.835590848428691, 4.106772661366034, 4.731368453025468, 5.055539243280185, 5.921236156021143, 6.399228511842473, 7.099634979558196, 7.562358016553593, 8.022819531163040, 8.142326514015190, 8.706477186735284, 9.485051078285522, 9.982158353597886, 10.52578680318593, 11.27664537686374, 11.41959534873894, 11.82683643347978, 12.27485780485367, 12.78549762842972, 13.24190212836186
