| 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 − 6·17-s + 18-s − 20-s − 21-s − 4·22-s − 8·23-s + 24-s + 25-s + 2·26-s + 27-s − 28-s − 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-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 − 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.993122809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.993122809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.89293907144860, −13.66819180958431, −13.05040841724218, −12.69250620638907, −12.28645149775864, −11.51518182977567, −11.11317236945385, −10.64395627982204, −10.12097868354643, −9.490871902474075, −8.968535247073823, −8.327045761538707, −7.858986202764069, −7.478047990542749, −6.828045511808433, −6.120619645879228, −5.873541269657000, −5.001102640777862, −4.451396178053970, −3.949995859931644, −3.461110801203180, −2.670905749687500, −2.302642090542788, −1.576730723225287, −0.3677304605337877,
0.3677304605337877, 1.576730723225287, 2.302642090542788, 2.670905749687500, 3.461110801203180, 3.949995859931644, 4.451396178053970, 5.001102640777862, 5.873541269657000, 6.120619645879228, 6.828045511808433, 7.478047990542749, 7.858986202764069, 8.327045761538707, 8.968535247073823, 9.490871902474075, 10.12097868354643, 10.64395627982204, 11.11317236945385, 11.51518182977567, 12.28645149775864, 12.69250620638907, 13.05040841724218, 13.66819180958431, 13.89293907144860
