| L(s) = 1 | − 0.212·2-s − 2.41·3-s − 1.95·4-s + 3.68·5-s + 0.513·6-s + 0.840·8-s + 2.83·9-s − 0.782·10-s + 4.70·11-s + 4.72·12-s + 2.04·13-s − 8.89·15-s + 3.73·16-s − 0.454·17-s − 0.602·18-s + 1.50·19-s − 7.20·20-s − 0.999·22-s + 3.25·23-s − 2.03·24-s + 8.56·25-s − 0.434·26-s + 0.392·27-s − 9.36·29-s + 1.89·30-s + 4.02·31-s − 2.47·32-s + ⋯ |
| L(s) = 1 | − 0.150·2-s − 1.39·3-s − 0.977·4-s + 1.64·5-s + 0.209·6-s + 0.297·8-s + 0.945·9-s − 0.247·10-s + 1.41·11-s + 1.36·12-s + 0.567·13-s − 2.29·15-s + 0.932·16-s − 0.110·17-s − 0.142·18-s + 0.345·19-s − 1.61·20-s − 0.213·22-s + 0.678·23-s − 0.414·24-s + 1.71·25-s − 0.0852·26-s + 0.0755·27-s − 1.73·29-s + 0.345·30-s + 0.723·31-s − 0.437·32-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
| \(L(1)\) |
\(\approx\) |
\(1.221223509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221223509\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.212T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 0.454T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 - 4.02T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 43 | \( 1 + 3.85T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 + 8.81T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 - 5.83T + 97T^{2} \) |
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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
−9.194343687597318147874615462489, −8.828240942471921310202671439025, −7.40455477755008188957437474377, −6.36912160914072394441634645956, −6.01192102451569379391320071537, −5.26801770948974075274650423978, −4.58086278965739441799319697775, −3.46470159252632477362721224210, −1.72955542198260597214900021996, −0.886925668032828437054197239680,
0.886925668032828437054197239680, 1.72955542198260597214900021996, 3.46470159252632477362721224210, 4.58086278965739441799319697775, 5.26801770948974075274650423978, 6.01192102451569379391320071537, 6.36912160914072394441634645956, 7.40455477755008188957437474377, 8.828240942471921310202671439025, 9.194343687597318147874615462489
