| L(s) = 1 | + 2.40·2-s + 0.306·3-s + 3.79·4-s + 5-s + 0.738·6-s + 3.95·7-s + 4.32·8-s − 2.90·9-s + 2.40·10-s − 2.21·11-s + 1.16·12-s + 6.68·13-s + 9.51·14-s + 0.306·15-s + 2.81·16-s + 0.577·17-s − 6.99·18-s + 3.79·20-s + 1.21·21-s − 5.34·22-s − 5.93·23-s + 1.32·24-s + 25-s + 16.1·26-s − 1.81·27-s + 14.9·28-s − 1.26·29-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.177·3-s + 1.89·4-s + 0.447·5-s + 0.301·6-s + 1.49·7-s + 1.52·8-s − 0.968·9-s + 0.761·10-s − 0.668·11-s + 0.336·12-s + 1.85·13-s + 2.54·14-s + 0.0792·15-s + 0.704·16-s + 0.140·17-s − 1.64·18-s + 0.848·20-s + 0.264·21-s − 1.13·22-s − 1.23·23-s + 0.270·24-s + 0.200·25-s + 3.15·26-s − 0.348·27-s + 2.83·28-s − 0.234·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.955910518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.955910518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 - 0.306T + 3T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 - 0.577T + 17T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 + 7.84T + 37T^{2} \) |
| 41 | \( 1 + 0.477T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.507T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 2.03T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 6.72T + 97T^{2} \) |
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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
−9.062444133705694149209807164108, −8.318487537034955952096873939426, −7.67078316679088560334014979744, −6.46493446291528434641567049899, −5.65337765942359848133525299010, −5.38640931157040820482624745743, −4.30161055047333588415891508885, −3.55760356068957442380836768360, −2.49573012615754140461700651993, −1.62088974650985597449960157731,
1.62088974650985597449960157731, 2.49573012615754140461700651993, 3.55760356068957442380836768360, 4.30161055047333588415891508885, 5.38640931157040820482624745743, 5.65337765942359848133525299010, 6.46493446291528434641567049899, 7.67078316679088560334014979744, 8.318487537034955952096873939426, 9.062444133705694149209807164108
