L(s) = 1 | − 2.54·2-s − 1.76·3-s + 4.48·4-s − 0.663·5-s + 4.50·6-s − 3.59·7-s − 6.32·8-s + 0.130·9-s + 1.69·10-s − 11-s − 7.93·12-s + 2.37·13-s + 9.15·14-s + 1.17·15-s + 7.13·16-s + 0.864·17-s − 0.331·18-s − 0.433·19-s − 2.97·20-s + 6.36·21-s + 2.54·22-s + 7.71·23-s + 11.1·24-s − 4.55·25-s − 6.03·26-s + 5.07·27-s − 16.1·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s − 1.02·3-s + 2.24·4-s − 0.296·5-s + 1.83·6-s − 1.35·7-s − 2.23·8-s + 0.0434·9-s + 0.534·10-s − 0.301·11-s − 2.28·12-s + 0.657·13-s + 2.44·14-s + 0.303·15-s + 1.78·16-s + 0.209·17-s − 0.0781·18-s − 0.0994·19-s − 0.665·20-s + 1.38·21-s + 0.542·22-s + 1.60·23-s + 2.28·24-s − 0.911·25-s − 1.18·26-s + 0.977·27-s − 3.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2763301333\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2763301333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 + 0.663T + 5T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 - 0.864T + 17T^{2} \) |
| 19 | \( 1 + 0.433T + 19T^{2} \) |
| 23 | \( 1 - 7.71T + 23T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 + 0.827T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 - 0.411T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.167311987084143766986700446370, −7.44075687715382884516888628293, −6.67104252147595295457300791480, −6.28994257285035204765821992774, −5.65381802416159911497164550551, −4.52149724123036819191496151868, −3.22523545323149167435699627294, −2.67445236486131823811975423984, −1.22791886139302142947363918681, −0.43367455259968497270600438251,
0.43367455259968497270600438251, 1.22791886139302142947363918681, 2.67445236486131823811975423984, 3.22523545323149167435699627294, 4.52149724123036819191496151868, 5.65381802416159911497164550551, 6.28994257285035204765821992774, 6.67104252147595295457300791480, 7.44075687715382884516888628293, 8.167311987084143766986700446370