| L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s + 13-s + 15-s + 17-s + 8·19-s + 25-s + 27-s + 6·29-s + 8·31-s + 4·33-s − 2·37-s + 39-s − 10·41-s + 4·43-s + 45-s − 7·49-s + 51-s + 6·53-s + 4·55-s + 8·57-s + 6·61-s + 65-s + 4·67-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s + 1.83·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.140·51-s + 0.824·53-s + 0.539·55-s + 1.05·57-s + 0.768·61-s + 0.124·65-s + 0.488·67-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.841337705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.841337705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.13169502913369, −12.49518078477079, −12.05570637324085, −11.54819721296557, −11.42693106504364, −10.41442511173711, −10.13131761248201, −9.728960623987710, −9.245557021410436, −8.815170797755247, −8.250733637776196, −7.965245490150759, −7.120149958876052, −6.853560845690891, −6.386043574732377, −5.736470530086847, −5.243459774198142, −4.679799287416098, −4.089197744047134, −3.536745173083963, −3.000147474183380, −2.598809679437062, −1.647800621580850, −1.283962475705765, −0.6994153242376597,
0.6994153242376597, 1.283962475705765, 1.647800621580850, 2.598809679437062, 3.000147474183380, 3.536745173083963, 4.089197744047134, 4.679799287416098, 5.243459774198142, 5.736470530086847, 6.386043574732377, 6.853560845690891, 7.120149958876052, 7.965245490150759, 8.250733637776196, 8.815170797755247, 9.245557021410436, 9.728960623987710, 10.13131761248201, 10.41442511173711, 11.42693106504364, 11.54819721296557, 12.05570637324085, 12.49518078477079, 13.13169502913369
