| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 2·13-s − 14-s + 16-s + 6·17-s + 18-s + 5·19-s − 21-s − 22-s + 23-s + 24-s + 2·26-s + 27-s − 28-s + 8·29-s + 4·31-s + 32-s − 33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.14·19-s − 0.218·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.718·31-s + 0.176·32-s − 0.174·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.603376698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.603376698\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.40259356323090, −14.78687974399123, −14.23946857452361, −13.81061913696558, −13.39654337004202, −12.82181637234835, −12.14726652401371, −11.87972704160650, −11.15845611209504, −10.35474540492147, −10.04282758056863, −9.450204434807167, −8.719707465347674, −7.991696029777103, −7.725570267142127, −6.883106640040863, −6.387942389455786, −5.659392844841057, −5.110660265843075, −4.423066374409473, −3.637697387034054, −3.065251312333451, −2.708973426093632, −1.539967017773856, −0.8762690625688995,
0.8762690625688995, 1.539967017773856, 2.708973426093632, 3.065251312333451, 3.637697387034054, 4.423066374409473, 5.110660265843075, 5.659392844841057, 6.387942389455786, 6.883106640040863, 7.725570267142127, 7.991696029777103, 8.719707465347674, 9.450204434807167, 10.04282758056863, 10.35474540492147, 11.15845611209504, 11.87972704160650, 12.14726652401371, 12.82181637234835, 13.39654337004202, 13.81061913696558, 14.23946857452361, 14.78687974399123, 15.40259356323090
