| L(s) = 1 | − 0.732·2-s − 3-s − 1.46·4-s − 0.267·5-s + 0.732·6-s − 7-s + 2.53·8-s + 9-s + 0.196·10-s + 1.46·12-s + 2.73·13-s + 0.732·14-s + 0.267·15-s + 1.07·16-s − 3.73·17-s − 0.732·18-s − 2.19·19-s + 0.392·20-s + 21-s + 3.26·23-s − 2.53·24-s − 4.92·25-s − 2·26-s − 27-s + 1.46·28-s + 1.26·29-s − 0.196·30-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.577·3-s − 0.732·4-s − 0.119·5-s + 0.298·6-s − 0.377·7-s + 0.896·8-s + 0.333·9-s + 0.0620·10-s + 0.422·12-s + 0.757·13-s + 0.195·14-s + 0.0691·15-s + 0.267·16-s − 0.905·17-s − 0.172·18-s − 0.503·19-s + 0.0877·20-s + 0.218·21-s + 0.681·23-s − 0.517·24-s − 0.985·25-s − 0.392·26-s − 0.192·27-s + 0.276·28-s + 0.235·29-s − 0.0358·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6388013511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6388013511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 5 | \( 1 + 0.267T + 5T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 0.464T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 9.12T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 + 0.535T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−8.806192853137549481826411549447, −8.381464303091072692591087458863, −7.41970180623759264440963663530, −6.59228837096630877548442198614, −5.88442055714005914450239116438, −4.86950747774797776140969758461, −4.24894808291035543973418875132, −3.30278121518021525712174392459, −1.80774748779950713604646221011, −0.57191706615010549654599890253,
0.57191706615010549654599890253, 1.80774748779950713604646221011, 3.30278121518021525712174392459, 4.24894808291035543973418875132, 4.86950747774797776140969758461, 5.88442055714005914450239116438, 6.59228837096630877548442198614, 7.41970180623759264440963663530, 8.381464303091072692591087458863, 8.806192853137549481826411549447
