L(s) = 1 | − 2.76·2-s − 1.76·3-s + 5.62·4-s + 2.62·5-s + 4.86·6-s − 10.0·8-s + 0.103·9-s − 7.25·10-s − 9.91·12-s − 2.38·13-s − 4.62·15-s + 16.4·16-s − 2.38·17-s − 0.284·18-s − 1.72·19-s + 14.7·20-s − 0.626·23-s + 17.6·24-s + 1.89·25-s + 6.59·26-s + 5.10·27-s − 1.72·29-s + 12.7·30-s + 2.23·31-s − 25.2·32-s + 6.59·34-s + 0.579·36-s + ⋯ |
L(s) = 1 | − 1.95·2-s − 1.01·3-s + 2.81·4-s + 1.17·5-s + 1.98·6-s − 3.54·8-s + 0.0343·9-s − 2.29·10-s − 2.86·12-s − 0.662·13-s − 1.19·15-s + 4.10·16-s − 0.579·17-s − 0.0670·18-s − 0.396·19-s + 3.30·20-s − 0.130·23-s + 3.60·24-s + 0.379·25-s + 1.29·26-s + 0.982·27-s − 0.321·29-s + 2.33·30-s + 0.402·31-s − 4.46·32-s + 1.13·34-s + 0.0966·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3823427144\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3823427144\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 - 8.08T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 8.35T + 97T^{2} \) |
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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
−8.207924557514166598038378322491, −7.45609783669711060151336012828, −6.60866703987631687593349306061, −6.28981654156943420911134746088, −5.63215216965030861164973721922, −4.80595727983828368423076209198, −3.15038888554770456422657705210, −2.24528511081675814040599468592, −1.62347841224271599684419824872, −0.45381627401680014378612570279,
0.45381627401680014378612570279, 1.62347841224271599684419824872, 2.24528511081675814040599468592, 3.15038888554770456422657705210, 4.80595727983828368423076209198, 5.63215216965030861164973721922, 6.28981654156943420911134746088, 6.60866703987631687593349306061, 7.45609783669711060151336012828, 8.207924557514166598038378322491