| L(s) = 1 | + 2.36·2-s − 1.64·3-s + 3.61·4-s − 4.02·5-s − 3.90·6-s + 3.82·8-s − 0.289·9-s − 9.54·10-s − 2.19·11-s − 5.95·12-s + 2.49·13-s + 6.63·15-s + 1.83·16-s + 4.73·17-s − 0.686·18-s + 0.560·19-s − 14.5·20-s − 5.21·22-s + 7.86·23-s − 6.29·24-s + 11.2·25-s + 5.90·26-s + 5.41·27-s + 1.33·29-s + 15.7·30-s + 4.18·31-s − 3.30·32-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.950·3-s + 1.80·4-s − 1.80·5-s − 1.59·6-s + 1.35·8-s − 0.0965·9-s − 3.01·10-s − 0.663·11-s − 1.71·12-s + 0.691·13-s + 1.71·15-s + 0.458·16-s + 1.14·17-s − 0.161·18-s + 0.128·19-s − 3.25·20-s − 1.11·22-s + 1.64·23-s − 1.28·24-s + 2.24·25-s + 1.15·26-s + 1.04·27-s + 0.247·29-s + 2.87·30-s + 0.752·31-s − 0.583·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.250367271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250367271\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 0.560T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 - 3.19T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 6.40T + 61T^{2} \) |
| 67 | \( 1 - 0.699T + 67T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 - 6.06T + 73T^{2} \) |
| 79 | \( 1 - 0.250T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 - 9.96T + 97T^{2} \) |
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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
−9.015205919090108018164308301105, −7.952001336390669646401897534841, −7.44213707370498665643409005316, −6.46070822548856247192881512375, −5.83365229844583297746203845503, −4.84043830206069221039106536861, −4.55939069977168282120351950497, −3.34494539504869939008120689598, −3.00014125008972791467771065151, −0.814161259989825622239246110483,
0.814161259989825622239246110483, 3.00014125008972791467771065151, 3.34494539504869939008120689598, 4.55939069977168282120351950497, 4.84043830206069221039106536861, 5.83365229844583297746203845503, 6.46070822548856247192881512375, 7.44213707370498665643409005316, 7.952001336390669646401897534841, 9.015205919090108018164308301105
