| L(s) = 1 | + 2-s − 4-s − 5-s − 7-s − 3·8-s − 10-s − 2·13-s − 14-s − 16-s + 20-s + 4·23-s + 25-s − 2·26-s + 28-s − 6·29-s + 5·32-s + 35-s + 2·37-s + 3·40-s + 6·41-s − 12·43-s + 4·46-s − 8·47-s + 49-s + 50-s + 2·52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 0.316·10-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.883·32-s + 0.169·35-s + 0.328·37-s + 0.474·40-s + 0.937·41-s − 1.82·43-s + 0.589·46-s − 1.16·47-s + 1/7·49-s + 0.141·50-s + 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7454040487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7454040487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.76502197495110, −13.30136363495982, −12.84810605513958, −12.62122842357744, −11.97662975739789, −11.41177102052149, −11.17734424857074, −10.32321187634016, −9.695684872653549, −9.570832189874420, −8.721042187562054, −8.489098543696708, −7.759620845921094, −7.182307674860147, −6.727249970302670, −6.040324793912742, −5.531438867336907, −5.002179959814475, −4.461061852739905, −4.010533936326442, −3.196880207221940, −3.063040679587815, −2.131443417622086, −1.229083836312360, −0.2574801791657925,
0.2574801791657925, 1.229083836312360, 2.131443417622086, 3.063040679587815, 3.196880207221940, 4.010533936326442, 4.461061852739905, 5.002179959814475, 5.531438867336907, 6.040324793912742, 6.727249970302670, 7.182307674860147, 7.759620845921094, 8.489098543696708, 8.721042187562054, 9.570832189874420, 9.695684872653549, 10.32321187634016, 11.17734424857074, 11.41177102052149, 11.97662975739789, 12.62122842357744, 12.84810605513958, 13.30136363495982, 13.76502197495110
