| L(s) = 1 | + 1.48·2-s + 1.67·3-s + 0.193·4-s + 2.48·6-s + 1.28·7-s − 2.67·8-s − 0.193·9-s − 0.481·11-s + 0.324·12-s + 2.15·13-s + 1.90·14-s − 4.35·16-s − 0.287·18-s − 3.35·19-s + 2.15·21-s − 0.712·22-s − 8.24·23-s − 4.48·24-s + 3.19·26-s − 5.35·27-s + 0.249·28-s − 0.649·29-s − 1.83·31-s − 1.09·32-s − 0.806·33-s − 0.0376·36-s + 4.31·37-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.967·3-s + 0.0969·4-s + 1.01·6-s + 0.486·7-s − 0.945·8-s − 0.0646·9-s − 0.145·11-s + 0.0937·12-s + 0.598·13-s + 0.509·14-s − 1.08·16-s − 0.0677·18-s − 0.768·19-s + 0.470·21-s − 0.151·22-s − 1.72·23-s − 0.914·24-s + 0.626·26-s − 1.02·27-s + 0.0471·28-s − 0.120·29-s − 0.328·31-s − 0.193·32-s − 0.140·33-s − 0.00626·36-s + 0.708·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 0.481T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 + 0.649T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8.15T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 + 4.93T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 3.66T + 97T^{2} \) |
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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
−7.79735833372565792864905755437, −6.70054531931396002341979540181, −5.96682764989111067632689206705, −5.46891691289771136122016079469, −4.46960559671611935088178275268, −3.99804831141305090356522148975, −3.28718335407292617876771505873, −2.51097472273898202800578433628, −1.71854695616944762347455401873, 0,
1.71854695616944762347455401873, 2.51097472273898202800578433628, 3.28718335407292617876771505873, 3.99804831141305090356522148975, 4.46960559671611935088178275268, 5.46891691289771136122016079469, 5.96682764989111067632689206705, 6.70054531931396002341979540181, 7.79735833372565792864905755437
