| L(s) = 1 | − 1.48·2-s + 0.806·3-s + 0.193·4-s + 3.15·5-s − 1.19·6-s + 0.675·7-s + 2.67·8-s − 2.35·9-s − 4.67·10-s + 2.51·11-s + 0.156·12-s − 6.35·13-s − 14-s + 2.54·15-s − 4.35·16-s − 4.15·17-s + 3.48·18-s − 0.418·19-s + 0.612·20-s + 0.544·21-s − 3.73·22-s + 3.48·23-s + 2.15·24-s + 4.96·25-s + 9.40·26-s − 4.31·27-s + 0.130·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.465·3-s + 0.0969·4-s + 1.41·5-s − 0.487·6-s + 0.255·7-s + 0.945·8-s − 0.783·9-s − 1.47·10-s + 0.759·11-s + 0.0451·12-s − 1.76·13-s − 0.267·14-s + 0.656·15-s − 1.08·16-s − 1.00·17-s + 0.820·18-s − 0.0959·19-s + 0.136·20-s + 0.118·21-s − 0.795·22-s + 0.725·23-s + 0.440·24-s + 0.992·25-s + 1.84·26-s − 0.829·27-s + 0.0247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6707920767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6707920767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 - 0.806T + 3T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 7 | \( 1 - 0.675T + 7T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 + 6.35T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 + 0.418T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 - 15.2T + 59T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 3.22T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−14.80818074476872608703064148917, −14.06973400813522684107601992760, −13.08784591500517795037492903944, −11.37502040003902649409728867532, −9.941973882760487341597755345894, −9.344546946873166470263184370971, −8.331043809385276815054729206828, −6.77572219622963771699791466067, −5.00001310439645679527377361739, −2.21260394775417123958667719120,
2.21260394775417123958667719120, 5.00001310439645679527377361739, 6.77572219622963771699791466067, 8.331043809385276815054729206828, 9.344546946873166470263184370971, 9.941973882760487341597755345894, 11.37502040003902649409728867532, 13.08784591500517795037492903944, 14.06973400813522684107601992760, 14.80818074476872608703064148917
