| L(s) = 1 | − 2-s + 4-s + 0.603·5-s + 4.55·7-s − 8-s − 0.603·10-s − 1.88·11-s + 0.332·13-s − 4.55·14-s + 16-s + 1.68·17-s + 2.93·19-s + 0.603·20-s + 1.88·22-s + 3.43·23-s − 4.63·25-s − 0.332·26-s + 4.55·28-s + 1.00·29-s + 4.33·31-s − 32-s − 1.68·34-s + 2.74·35-s − 4.29·37-s − 2.93·38-s − 0.603·40-s − 3.48·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.269·5-s + 1.71·7-s − 0.353·8-s − 0.190·10-s − 0.568·11-s + 0.0923·13-s − 1.21·14-s + 0.250·16-s + 0.408·17-s + 0.672·19-s + 0.134·20-s + 0.401·22-s + 0.715·23-s − 0.927·25-s − 0.0652·26-s + 0.859·28-s + 0.186·29-s + 0.779·31-s − 0.176·32-s − 0.288·34-s + 0.463·35-s − 0.705·37-s − 0.475·38-s − 0.0953·40-s − 0.543·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.867041863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867041863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 0.603T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 - 0.332T + 13T^{2} \) |
| 17 | \( 1 - 1.68T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 37 | \( 1 + 4.29T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 5.00T + 61T^{2} \) |
| 67 | \( 1 + 0.731T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−8.330939590452743176592296970522, −7.84810590611968803541882772632, −7.28828946567960961879212697518, −6.30110038591605800043111663813, −5.33510548696956114466463192575, −4.95260826231655737005185476631, −3.81115090377158829079594941246, −2.65413020299433357294859817626, −1.80698609402567078853501296222, −0.923891473379773069137066170802,
0.923891473379773069137066170802, 1.80698609402567078853501296222, 2.65413020299433357294859817626, 3.81115090377158829079594941246, 4.95260826231655737005185476631, 5.33510548696956114466463192575, 6.30110038591605800043111663813, 7.28828946567960961879212697518, 7.84810590611968803541882772632, 8.330939590452743176592296970522
