| L(s) = 1 | − 1.19·2-s − 3-s − 0.561·4-s − 0.150·5-s + 1.19·6-s − 7-s + 3.07·8-s + 9-s + 0.180·10-s − 3.04·11-s + 0.561·12-s + 2.11·13-s + 1.19·14-s + 0.150·15-s − 2.56·16-s − 7.20·17-s − 1.19·18-s − 2.33·19-s + 0.0842·20-s + 21-s + 3.65·22-s − 7.27·23-s − 3.07·24-s − 4.97·25-s − 2.53·26-s − 27-s + 0.561·28-s + ⋯ |
| L(s) = 1 | − 0.848·2-s − 0.577·3-s − 0.280·4-s − 0.0671·5-s + 0.489·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s + 0.0569·10-s − 0.919·11-s + 0.162·12-s + 0.586·13-s + 0.320·14-s + 0.0387·15-s − 0.640·16-s − 1.74·17-s − 0.282·18-s − 0.536·19-s + 0.0188·20-s + 0.218·21-s + 0.779·22-s − 1.51·23-s − 0.627·24-s − 0.995·25-s − 0.497·26-s − 0.192·27-s + 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.008623934918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008623934918\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 5 | \( 1 + 0.150T + 5T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 7.20T + 17T^{2} \) |
| 19 | \( 1 + 2.33T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 + 0.178T + 37T^{2} \) |
| 41 | \( 1 - 1.54T + 41T^{2} \) |
| 43 | \( 1 - 7.93T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + 5.59T + 59T^{2} \) |
| 61 | \( 1 + 3.48T + 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 - 6.07T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.78462351443987147970543735648, −7.42299597550566031409916921644, −6.39264655196372955104096800897, −5.90368718959242017361361773118, −5.05866994985717049527243161423, −4.22472034244657087129382998080, −3.76470020834594983263750601543, −2.31782611704567556041479894221, −1.64734614155862440374457282409, −0.05289803124649751564828623561,
0.05289803124649751564828623561, 1.64734614155862440374457282409, 2.31782611704567556041479894221, 3.76470020834594983263750601543, 4.22472034244657087129382998080, 5.05866994985717049527243161423, 5.90368718959242017361361773118, 6.39264655196372955104096800897, 7.42299597550566031409916921644, 7.78462351443987147970543735648
