| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 7-s − 3·8-s + 9-s + 10-s − 12-s + 2·13-s − 14-s + 15-s − 16-s − 2·17-s + 18-s − 4·19-s − 20-s − 21-s − 3·24-s + 25-s + 2·26-s + 27-s + 28-s − 6·29-s + 30-s + 5·32-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 + 0.554·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.182·30-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.838718943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838718943\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.04728841485029, −15.66018979772464, −14.92385294408248, −14.52813395303944, −14.07501704395503, −13.30031209940116, −13.06959769848525, −12.70875851458820, −11.88328333071177, −11.15746656559748, −10.54048119659661, −9.798476327047058, −9.218449282760321, −8.887626552192461, −8.184822994724091, −7.470192407245361, −6.575398018147698, −6.063928540332243, −5.505367665506410, −4.571798821012665, −4.107425908716360, −3.420678591751660, −2.663721987545108, −1.911021293358396, −0.6489490189786470,
0.6489490189786470, 1.911021293358396, 2.663721987545108, 3.420678591751660, 4.107425908716360, 4.571798821012665, 5.505367665506410, 6.063928540332243, 6.575398018147698, 7.470192407245361, 8.184822994724091, 8.887626552192461, 9.218449282760321, 9.798476327047058, 10.54048119659661, 11.15746656559748, 11.88328333071177, 12.70875851458820, 13.06959769848525, 13.30031209940116, 14.07501704395503, 14.52813395303944, 14.92385294408248, 15.66018979772464, 16.04728841485029
