| L(s) = 1 | + 2.09·2-s + 3-s + 2.40·4-s − 3.29·5-s + 2.09·6-s + 7-s + 0.858·8-s + 9-s − 6.90·10-s − 3.62·11-s + 2.40·12-s − 2.16·13-s + 2.09·14-s − 3.29·15-s − 3.01·16-s + 2.19·17-s + 2.09·18-s − 6.66·19-s − 7.92·20-s + 21-s − 7.61·22-s + 2.36·23-s + 0.858·24-s + 5.82·25-s − 4.53·26-s + 27-s + 2.40·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.20·4-s − 1.47·5-s + 0.857·6-s + 0.377·7-s + 0.303·8-s + 0.333·9-s − 2.18·10-s − 1.09·11-s + 0.695·12-s − 0.599·13-s + 0.561·14-s − 0.849·15-s − 0.753·16-s + 0.531·17-s + 0.494·18-s − 1.52·19-s − 1.77·20-s + 0.218·21-s − 1.62·22-s + 0.493·23-s + 0.175·24-s + 1.16·25-s − 0.889·26-s + 0.192·27-s + 0.455·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 + 6.66T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 0.647T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 8.59T + 47T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + 0.592T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 0.664T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 3.24T + 73T^{2} \) |
| 79 | \( 1 - 4.01T + 79T^{2} \) |
| 83 | \( 1 + 3.40T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 + 0.0689T + 97T^{2} \) |
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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
−8.330393042250060725274878315276, −7.55700246370403466597088486816, −7.04166882073594139322080390504, −5.97910815232961638707404265214, −4.88874203030223447246842407383, −4.59881785531194680942793839419, −3.64286368313824415727375722646, −3.06076565460433692811602790337, −2.08995788913651934432165487272, 0,
2.08995788913651934432165487272, 3.06076565460433692811602790337, 3.64286368313824415727375722646, 4.59881785531194680942793839419, 4.88874203030223447246842407383, 5.97910815232961638707404265214, 7.04166882073594139322080390504, 7.55700246370403466597088486816, 8.330393042250060725274878315276
