| L(s) = 1 | + 0.381·2-s + 2.23·3-s − 1.85·4-s − 5-s + 0.854·6-s − 7-s − 1.47·8-s + 2.00·9-s − 0.381·10-s + 6·11-s − 4.14·12-s + 3·13-s − 0.381·14-s − 2.23·15-s + 3.14·16-s − 1.23·17-s + 0.763·18-s + 8.47·19-s + 1.85·20-s − 2.23·21-s + 2.29·22-s + 23-s − 3.29·24-s + 25-s + 1.14·26-s − 2.23·27-s + 1.85·28-s + ⋯ |
| L(s) = 1 | + 0.270·2-s + 1.29·3-s − 0.927·4-s − 0.447·5-s + 0.348·6-s − 0.377·7-s − 0.520·8-s + 0.666·9-s − 0.120·10-s + 1.80·11-s − 1.19·12-s + 0.832·13-s − 0.102·14-s − 0.577·15-s + 0.786·16-s − 0.299·17-s + 0.180·18-s + 1.94·19-s + 0.414·20-s − 0.487·21-s + 0.488·22-s + 0.208·23-s − 0.671·24-s + 0.200·25-s + 0.224·26-s − 0.430·27-s + 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.190122870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.190122870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 8.47T + 19T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 + 4.23T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 + 0.291T + 97T^{2} \) |
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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
−9.811526707276010454910905576922, −9.185084765820485801202048715220, −8.790710241898100608148865149624, −7.912625264747817870861241762855, −6.91566101192373489755903822396, −5.82666458287829346172264643035, −4.51618064071827333276257497265, −3.61197277555420765132320641155, −3.17925766455958072218857924608, −1.24917249472925795318970977392,
1.24917249472925795318970977392, 3.17925766455958072218857924608, 3.61197277555420765132320641155, 4.51618064071827333276257497265, 5.82666458287829346172264643035, 6.91566101192373489755903822396, 7.912625264747817870861241762855, 8.790710241898100608148865149624, 9.185084765820485801202048715220, 9.811526707276010454910905576922
