| L(s) = 1 | − 2.22·3-s − 3.95·5-s + 3.08·7-s + 1.94·9-s + 3.50·11-s + 3.47·13-s + 8.79·15-s − 3.04·17-s − 7.32·19-s − 6.84·21-s + 6.16·23-s + 10.6·25-s + 2.35·27-s − 3.33·29-s − 5.01·31-s − 7.78·33-s − 12.1·35-s − 9.39·37-s − 7.71·39-s − 2.63·41-s − 0.457·43-s − 7.67·45-s + 4.87·47-s + 2.48·49-s + 6.78·51-s + 0.519·53-s − 13.8·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 1.76·5-s + 1.16·7-s + 0.647·9-s + 1.05·11-s + 0.963·13-s + 2.26·15-s − 0.739·17-s − 1.68·19-s − 1.49·21-s + 1.28·23-s + 2.12·25-s + 0.452·27-s − 0.619·29-s − 0.900·31-s − 1.35·33-s − 2.05·35-s − 1.54·37-s − 1.23·39-s − 0.412·41-s − 0.0696·43-s − 1.14·45-s + 0.711·47-s + 0.355·49-s + 0.949·51-s + 0.0714·53-s − 1.86·55-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 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 7.32T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 + 3.33T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 + 2.63T + 41T^{2} \) |
| 43 | \( 1 + 0.457T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 0.519T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.71T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.36T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 7.25T + 97T^{2} \) |
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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
−7.28383826545137194513293192201, −6.73115812837997152850854487407, −6.17152035850038834233123120376, −5.12220625367859068007454219904, −4.68307411370621162025373054446, −3.99138918303333192204399913679, −3.46578499520248589917252880591, −1.91486566345277686654663973470, −0.963024184196562092166983926331, 0,
0.963024184196562092166983926331, 1.91486566345277686654663973470, 3.46578499520248589917252880591, 3.99138918303333192204399913679, 4.68307411370621162025373054446, 5.12220625367859068007454219904, 6.17152035850038834233123120376, 6.73115812837997152850854487407, 7.28383826545137194513293192201
