| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s − 12-s − 6·13-s + 14-s + 15-s − 16-s + 2·17-s + 18-s − 8·19-s − 20-s + 21-s + 8·23-s − 3·24-s + 25-s − 6·26-s + 27-s − 28-s − 2·29-s + 30-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 − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.465863363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465863363\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.78714664919275649975914854040, −12.89916852888262332222898282203, −12.11417775373724942408035215331, −10.47975989489334108877507822892, −9.386500711260631787757553076630, −8.447489066361171687346291758504, −6.97944382340517249810347443111, −5.37573342552736399524296425305, −4.33309058948755389799390879771, −2.63294756479994443235170715895,
2.63294756479994443235170715895, 4.33309058948755389799390879771, 5.37573342552736399524296425305, 6.97944382340517249810347443111, 8.447489066361171687346291758504, 9.386500711260631787757553076630, 10.47975989489334108877507822892, 12.11417775373724942408035215331, 12.89916852888262332222898282203, 13.78714664919275649975914854040
