| L(s) = 1 | − 1.70·2-s + 3-s + 0.916·4-s + 2.36·5-s − 1.70·6-s + 7-s + 1.84·8-s + 9-s − 4.04·10-s − 0.746·11-s + 0.916·12-s + 1.43·13-s − 1.70·14-s + 2.36·15-s − 4.99·16-s − 0.954·17-s − 1.70·18-s + 0.503·19-s + 2.17·20-s + 21-s + 1.27·22-s + 3.97·23-s + 1.84·24-s + 0.602·25-s − 2.44·26-s + 27-s + 0.916·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.577·3-s + 0.458·4-s + 1.05·5-s − 0.697·6-s + 0.377·7-s + 0.654·8-s + 0.333·9-s − 1.27·10-s − 0.225·11-s + 0.264·12-s + 0.397·13-s − 0.456·14-s + 0.611·15-s − 1.24·16-s − 0.231·17-s − 0.402·18-s + 0.115·19-s + 0.485·20-s + 0.218·21-s + 0.271·22-s + 0.828·23-s + 0.377·24-s + 0.120·25-s − 0.480·26-s + 0.192·27-s + 0.173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.622010034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622010034\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 11 | \( 1 + 0.746T + 11T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 0.954T + 17T^{2} \) |
| 19 | \( 1 - 0.503T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 - 0.415T + 43T^{2} \) |
| 47 | \( 1 + 0.842T + 47T^{2} \) |
| 53 | \( 1 - 0.595T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 7.43T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 + 0.569T + 97T^{2} \) |
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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
−8.673371469029848497143838712944, −7.87664338705536940770269628302, −7.28009134778380668115939262111, −6.44616142177995940190774470283, −5.53544216307964594510245004102, −4.73212183162137787598255360795, −3.74799878448149444439154621610, −2.50713614929548459928288230273, −1.84273100534347960926731863459, −0.894824332434292263949434129620,
0.894824332434292263949434129620, 1.84273100534347960926731863459, 2.50713614929548459928288230273, 3.74799878448149444439154621610, 4.73212183162137787598255360795, 5.53544216307964594510245004102, 6.44616142177995940190774470283, 7.28009134778380668115939262111, 7.87664338705536940770269628302, 8.673371469029848497143838712944
