| L(s) = 1 | + 8.48·5-s + 7·7-s + 25.4·13-s + 8.48·19-s − 10·23-s + 46.9·25-s + 59.3·35-s + 49·49-s − 76.3·59-s − 8.48·61-s + 215.·65-s − 110·71-s − 130·79-s + 25.4·83-s + 178.·91-s + 71.9·95-s − 161.·101-s + 26·113-s − 84.8·115-s + ⋯ |
| L(s) = 1 | + 1.69·5-s + 7-s + 1.95·13-s + 0.446·19-s − 0.434·23-s + 1.87·25-s + 1.69·35-s + 0.999·49-s − 1.29·59-s − 0.139·61-s + 3.32·65-s − 1.54·71-s − 1.64·79-s + 0.306·83-s + 1.95·91-s + 0.757·95-s − 1.59·101-s + 0.230·113-s − 0.737·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.847943847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.847943847\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 8.48T + 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 25.4T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 8.48T + 361T^{2} \) |
| 23 | \( 1 + 10T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 76.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 8.48T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 110T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 130T + 6.24e3T^{2} \) |
| 83 | \( 1 - 25.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−8.905413955180413903731995308448, −8.417610953695573731439594829375, −7.41049543057399357203140370751, −6.31084852570963543737342631501, −5.86964000995592944723267348662, −5.12633117194320711491560355762, −4.10487603039135096539346078530, −2.91757824631925370332415185022, −1.76707012980805688642068785522, −1.22436656333937190535166331034,
1.22436656333937190535166331034, 1.76707012980805688642068785522, 2.91757824631925370332415185022, 4.10487603039135096539346078530, 5.12633117194320711491560355762, 5.86964000995592944723267348662, 6.31084852570963543737342631501, 7.41049543057399357203140370751, 8.417610953695573731439594829375, 8.905413955180413903731995308448
