L(s) = 1 | − 2.60·2-s + 0.316·3-s + 4.77·4-s + 1.57·5-s − 0.822·6-s + 2.80·7-s − 7.23·8-s − 2.90·9-s − 4.10·10-s + 1.05·11-s + 1.51·12-s + 1.29·13-s − 7.29·14-s + 0.498·15-s + 9.28·16-s + 2.11·17-s + 7.55·18-s + 4.99·19-s + 7.54·20-s + 0.884·21-s − 2.74·22-s + 7.80·23-s − 2.28·24-s − 2.50·25-s − 3.36·26-s − 1.86·27-s + 13.3·28-s + ⋯ |
L(s) = 1 | − 1.84·2-s + 0.182·3-s + 2.38·4-s + 0.705·5-s − 0.335·6-s + 1.05·7-s − 2.55·8-s − 0.966·9-s − 1.29·10-s + 0.317·11-s + 0.436·12-s + 0.358·13-s − 1.94·14-s + 0.128·15-s + 2.32·16-s + 0.514·17-s + 1.77·18-s + 1.14·19-s + 1.68·20-s + 0.193·21-s − 0.584·22-s + 1.62·23-s − 0.466·24-s − 0.501·25-s − 0.659·26-s − 0.358·27-s + 2.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.216458304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216458304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4003 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 0.316T + 3T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 - 2.11T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 + 6.15T + 31T^{2} \) |
| 37 | \( 1 - 7.71T + 37T^{2} \) |
| 41 | \( 1 - 2.40T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 7.78T + 53T^{2} \) |
| 59 | \( 1 - 9.71T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 9.20T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 + 5.44T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−8.459975913709573949710199388915, −7.990906671897542642513444646161, −7.30833500282404400974621922317, −6.48173950586031679947803703966, −5.70009564693066351296536061964, −4.98056073942278911254697530330, −3.36637840655700901821735649404, −2.55890292543467457058434313022, −1.62091772960557621474087076035, −0.888887932937329406339411219593,
0.888887932937329406339411219593, 1.62091772960557621474087076035, 2.55890292543467457058434313022, 3.36637840655700901821735649404, 4.98056073942278911254697530330, 5.70009564693066351296536061964, 6.48173950586031679947803703966, 7.30833500282404400974621922317, 7.990906671897542642513444646161, 8.459975913709573949710199388915