| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s − 4·11-s + 13-s − 3·15-s − 7·17-s + 19-s − 3·21-s + 5·23-s + 25-s − 9·27-s + 7·29-s + 2·31-s + 12·33-s + 35-s − 6·37-s − 3·39-s + 6·41-s − 10·43-s + 6·45-s + 8·47-s − 6·49-s + 21·51-s − 3·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.20·11-s + 0.277·13-s − 0.774·15-s − 1.69·17-s + 0.229·19-s − 0.654·21-s + 1.04·23-s + 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.359·31-s + 2.08·33-s + 0.169·35-s − 0.986·37-s − 0.480·39-s + 0.937·41-s − 1.52·43-s + 0.894·45-s + 1.16·47-s − 6/7·49-s + 2.94·51-s − 0.412·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.166794634373524225750658480305, −8.244607052792420796753047387180, −7.13143932663839007991861548890, −6.53248965558733217968462373412, −5.72947607652733769777621747465, −4.94911729364798335949026706022, −4.48492827514281023815695123179, −2.76829964334084701202004565949, −1.41596078744357748965463468505, 0,
1.41596078744357748965463468505, 2.76829964334084701202004565949, 4.48492827514281023815695123179, 4.94911729364798335949026706022, 5.72947607652733769777621747465, 6.53248965558733217968462373412, 7.13143932663839007991861548890, 8.244607052792420796753047387180, 9.166794634373524225750658480305
