| L(s) = 1 | + 2-s + 1.17·3-s + 4-s + 1.17·6-s − 2.08·7-s + 8-s − 1.61·9-s + 2.99·11-s + 1.17·12-s − 2.08·14-s + 16-s + 1.26·17-s − 1.61·18-s + 3.89·19-s − 2.45·21-s + 2.99·22-s + 0.899·23-s + 1.17·24-s − 5.43·27-s − 2.08·28-s − 0.470·29-s + 0.277·31-s + 32-s + 3.52·33-s + 1.26·34-s − 1.61·36-s + 7.03·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.679·3-s + 0.5·4-s + 0.480·6-s − 0.788·7-s + 0.353·8-s − 0.538·9-s + 0.902·11-s + 0.339·12-s − 0.557·14-s + 0.250·16-s + 0.305·17-s − 0.380·18-s + 0.892·19-s − 0.535·21-s + 0.637·22-s + 0.187·23-s + 0.240·24-s − 1.04·27-s − 0.394·28-s − 0.0874·29-s + 0.0497·31-s + 0.176·32-s + 0.612·33-s + 0.216·34-s − 0.269·36-s + 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.001168255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.001168255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 - 0.899T + 23T^{2} \) |
| 29 | \( 1 + 0.470T + 29T^{2} \) |
| 31 | \( 1 - 0.277T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
| 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
−7.84992266744580808276987844535, −6.84289286056091034537751443254, −6.54608881706580421641782999089, −5.61794260643559970965012855469, −5.10503943460650738912634825047, −3.94519092527253347691110154099, −3.53965698729157575413896474177, −2.85527657220175220056295335903, −2.05890509569967555760889545293, −0.848821126778620410789214071315,
0.848821126778620410789214071315, 2.05890509569967555760889545293, 2.85527657220175220056295335903, 3.53965698729157575413896474177, 3.94519092527253347691110154099, 5.10503943460650738912634825047, 5.61794260643559970965012855469, 6.54608881706580421641782999089, 6.84289286056091034537751443254, 7.84992266744580808276987844535
