| L(s) = 1 | + 5·2-s − 5·3-s + 17·4-s + 5·5-s − 25·6-s − 19·7-s + 45·8-s − 2·9-s + 25·10-s + 50·11-s − 85·12-s − 55·13-s − 95·14-s − 25·15-s + 89·16-s + 51·17-s − 10·18-s + 85·20-s + 95·21-s + 250·22-s − 147·23-s − 225·24-s + 25·25-s − 275·26-s + 145·27-s − 323·28-s + 165·29-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.962·3-s + 17/8·4-s + 0.447·5-s − 1.70·6-s − 1.02·7-s + 1.98·8-s − 0.0740·9-s + 0.790·10-s + 1.37·11-s − 2.04·12-s − 1.17·13-s − 1.81·14-s − 0.430·15-s + 1.39·16-s + 0.727·17-s − 0.130·18-s + 0.950·20-s + 0.987·21-s + 2.42·22-s − 1.33·23-s − 1.91·24-s + 1/5·25-s − 2.07·26-s + 1.03·27-s − 2.18·28-s + 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 55 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 147 T + p^{3} T^{2} \) |
| 29 | \( 1 - 165 T + p^{3} T^{2} \) |
| 31 | \( 1 - 70 T + p^{3} T^{2} \) |
| 37 | \( 1 + 210 T + p^{3} T^{2} \) |
| 41 | \( 1 - 80 T + p^{3} T^{2} \) |
| 43 | \( 1 + 558 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 455 T + p^{3} T^{2} \) |
| 59 | \( 1 + 225 T + p^{3} T^{2} \) |
| 61 | \( 1 - 500 T + p^{3} T^{2} \) |
| 67 | \( 1 + 105 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1140 T + p^{3} T^{2} \) |
| 73 | \( 1 + 703 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 + 918 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1380 T + p^{3} T^{2} \) |
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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
−8.401976794192229263600334832157, −7.03237397308165420790572925005, −6.49303272032341418322698860223, −6.07290648823395753890733613321, −5.23911343469264166133355962373, −4.57727763048492438987663814157, −3.54976049675451558474325601514, −2.82775283888015748129448844055, −1.59332871488771549564304412296, 0,
1.59332871488771549564304412296, 2.82775283888015748129448844055, 3.54976049675451558474325601514, 4.57727763048492438987663814157, 5.23911343469264166133355962373, 6.07290648823395753890733613321, 6.49303272032341418322698860223, 7.03237397308165420790572925005, 8.401976794192229263600334832157
