| L(s) = 1 | − 0.384·2-s − 0.221·3-s − 1.85·4-s + 1.35·5-s + 0.0850·6-s − 3.71·7-s + 1.47·8-s − 2.95·9-s − 0.519·10-s + 11-s + 0.410·12-s − 2.48·13-s + 1.42·14-s − 0.299·15-s + 3.13·16-s + 17-s + 1.13·18-s − 8.36·19-s − 2.50·20-s + 0.821·21-s − 0.384·22-s − 4.42·23-s − 0.327·24-s − 3.17·25-s + 0.955·26-s + 1.31·27-s + 6.87·28-s + ⋯ |
| L(s) = 1 | − 0.271·2-s − 0.127·3-s − 0.926·4-s + 0.604·5-s + 0.0347·6-s − 1.40·7-s + 0.523·8-s − 0.983·9-s − 0.164·10-s + 0.301·11-s + 0.118·12-s − 0.690·13-s + 0.380·14-s − 0.0773·15-s + 0.784·16-s + 0.242·17-s + 0.267·18-s − 1.91·19-s − 0.560·20-s + 0.179·21-s − 0.0818·22-s − 0.921·23-s − 0.0668·24-s − 0.634·25-s + 0.187·26-s + 0.253·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1022939750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1022939750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.384T + 2T^{2} \) |
| 3 | \( 1 + 0.221T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 19 | \( 1 + 8.36T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 + 0.657T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 47 | \( 1 + 2.05T + 47T^{2} \) |
| 53 | \( 1 - 0.410T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 8.64T + 73T^{2} \) |
| 79 | \( 1 + 0.538T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.090457222879915432562626364836, −6.97491671874984006767408269261, −6.44205704938765083729362638892, −5.71038515226673640103437349095, −5.24192892294935784925471956581, −4.14828996240316875286449548062, −3.61440236241771199195701490428, −2.65048012462281030754191236531, −1.77180576023637281614961621636, −0.15843677687734859124549775770,
0.15843677687734859124549775770, 1.77180576023637281614961621636, 2.65048012462281030754191236531, 3.61440236241771199195701490428, 4.14828996240316875286449548062, 5.24192892294935784925471956581, 5.71038515226673640103437349095, 6.44205704938765083729362638892, 6.97491671874984006767408269261, 8.090457222879915432562626364836
