| L(s) = 1 | + 2.71·3-s + 0.714·5-s + 2.03·7-s + 4.38·9-s − 5.43·11-s − 3.96·13-s + 1.94·15-s − 1.30·17-s − 8.24·19-s + 5.52·21-s + 2.44·23-s − 4.49·25-s + 3.76·27-s + 1.49·29-s + 6.02·31-s − 14.7·33-s + 1.45·35-s − 6.36·37-s − 10.7·39-s − 2.87·41-s + 5.76·43-s + 3.13·45-s − 6.42·47-s − 2.86·49-s − 3.55·51-s − 0.932·53-s − 3.88·55-s + ⋯ |
| L(s) = 1 | + 1.56·3-s + 0.319·5-s + 0.768·7-s + 1.46·9-s − 1.63·11-s − 1.10·13-s + 0.501·15-s − 0.317·17-s − 1.89·19-s + 1.20·21-s + 0.510·23-s − 0.898·25-s + 0.725·27-s + 0.277·29-s + 1.08·31-s − 2.57·33-s + 0.245·35-s − 1.04·37-s − 1.72·39-s − 0.448·41-s + 0.879·43-s + 0.466·45-s − 0.937·47-s − 0.409·49-s − 0.497·51-s − 0.128·53-s − 0.523·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 - 0.714T + 5T^{2} \) |
| 7 | \( 1 - 2.03T + 7T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 + 6.36T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 + 0.932T + 53T^{2} \) |
| 59 | \( 1 + 1.57T + 59T^{2} \) |
| 61 | \( 1 + 7.93T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 9.81T + 73T^{2} \) |
| 79 | \( 1 - 2.80T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 - 0.276T + 97T^{2} \) |
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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
−7.71118404661727666933671624926, −6.96603945451797345838880386985, −6.11914292963630249602403532735, −4.99736087809536418564790326409, −4.71431452087254352887379122559, −3.76029488019305579113421360955, −2.76263732268514731457800454597, −2.34436458838405588571250472927, −1.74172153523115714109997958377, 0,
1.74172153523115714109997958377, 2.34436458838405588571250472927, 2.76263732268514731457800454597, 3.76029488019305579113421360955, 4.71431452087254352887379122559, 4.99736087809536418564790326409, 6.11914292963630249602403532735, 6.96603945451797345838880386985, 7.71118404661727666933671624926
