L(s) = 1 | + 2·5-s − 2·11-s − 2·13-s + 6·17-s + 4·19-s − 6·23-s − 25-s + 4·31-s + 10·37-s + 2·41-s − 4·43-s + 4·47-s + 12·53-s − 4·55-s + 12·59-s − 6·61-s − 4·65-s − 4·67-s + 14·71-s + 2·73-s − 8·79-s − 16·83-s + 12·85-s − 6·89-s + 8·95-s + 18·97-s + 14·101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.603·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 1/5·25-s + 0.718·31-s + 1.64·37-s + 0.312·41-s − 0.609·43-s + 0.583·47-s + 1.64·53-s − 0.539·55-s + 1.56·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s + 1.66·71-s + 0.234·73-s − 0.900·79-s − 1.75·83-s + 1.30·85-s − 0.635·89-s + 0.820·95-s + 1.82·97-s + 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.223761047\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.223761047\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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
−8.504793834320953202628646556657, −7.76848150678085187928223533831, −7.24329071872849616702174177955, −6.06736147014371865029363695465, −5.68448447926530784623104082980, −4.92527781611760469049753721027, −3.88751528788529186634819803030, −2.86097224409282808146377762278, −2.10144508111994341368241632876, −0.894513579519257675052133664689,
0.894513579519257675052133664689, 2.10144508111994341368241632876, 2.86097224409282808146377762278, 3.88751528788529186634819803030, 4.92527781611760469049753721027, 5.68448447926530784623104082980, 6.06736147014371865029363695465, 7.24329071872849616702174177955, 7.76848150678085187928223533831, 8.504793834320953202628646556657