| L(s) = 1 | − 1.53·2-s + 3.17·3-s + 0.369·4-s − 4.87·6-s + 1.70·7-s + 2.51·8-s + 7.04·9-s + 2.53·11-s + 1.17·12-s − 2.63·14-s − 4.60·16-s + 0.921·17-s − 10.8·18-s + 0.539·19-s + 5.41·21-s − 3.90·22-s + 2.82·23-s + 7.95·24-s + 12.8·27-s + 0.630·28-s − 5.12·29-s − 0.879·31-s + 2.06·32-s + 8.04·33-s − 1.41·34-s + 2.60·36-s + 6.04·37-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 1.83·3-s + 0.184·4-s − 1.99·6-s + 0.646·7-s + 0.887·8-s + 2.34·9-s + 0.765·11-s + 0.337·12-s − 0.703·14-s − 1.15·16-s + 0.223·17-s − 2.55·18-s + 0.123·19-s + 1.18·21-s − 0.833·22-s + 0.590·23-s + 1.62·24-s + 2.47·27-s + 0.119·28-s − 0.952·29-s − 0.157·31-s + 0.364·32-s + 1.40·33-s − 0.243·34-s + 0.433·36-s + 0.994·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.546539063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546539063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 17 | \( 1 - 0.921T + 17T^{2} \) |
| 19 | \( 1 - 0.539T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 0.879T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 8.49T + 53T^{2} \) |
| 59 | \( 1 - 4.72T + 59T^{2} \) |
| 61 | \( 1 - 8.04T + 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 - 0.496T + 79T^{2} \) |
| 83 | \( 1 + 8.63T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−8.440707980913021783870363700892, −7.963382331328677559287771993121, −7.35244973994674970684762225264, −6.73599524612931229802660795376, −5.30257276347012789444031664621, −4.29586704112664020415335260238, −3.78588152142301164415638594187, −2.69660391717478934282217608529, −1.81661022435050964775859050973, −1.08822136242145594411617037081,
1.08822136242145594411617037081, 1.81661022435050964775859050973, 2.69660391717478934282217608529, 3.78588152142301164415638594187, 4.29586704112664020415335260238, 5.30257276347012789444031664621, 6.73599524612931229802660795376, 7.35244973994674970684762225264, 7.963382331328677559287771993121, 8.440707980913021783870363700892
