L(s) = 1 | + 2-s + 2.06·3-s + 4-s − 2.29·5-s + 2.06·6-s + 0.0148·7-s + 8-s + 1.24·9-s − 2.29·10-s − 11-s + 2.06·12-s − 6.98·13-s + 0.0148·14-s − 4.72·15-s + 16-s + 2.20·17-s + 1.24·18-s + 3.95·19-s − 2.29·20-s + 0.0305·21-s − 22-s − 2.46·23-s + 2.06·24-s + 0.264·25-s − 6.98·26-s − 3.61·27-s + 0.0148·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s − 1.02·5-s + 0.841·6-s + 0.00560·7-s + 0.353·8-s + 0.414·9-s − 0.725·10-s − 0.301·11-s + 0.594·12-s − 1.93·13-s + 0.00396·14-s − 1.22·15-s + 0.250·16-s + 0.534·17-s + 0.293·18-s + 0.906·19-s − 0.513·20-s + 0.00666·21-s − 0.213·22-s − 0.514·23-s + 0.420·24-s + 0.0529·25-s − 1.36·26-s − 0.696·27-s + 0.00280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 - 0.0148T + 7T^{2} \) |
| 13 | \( 1 + 6.98T + 13T^{2} \) |
| 17 | \( 1 - 2.20T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 0.400T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + 0.740T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 + 7.19T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 - 5.80T + 89T^{2} \) |
| 97 | \( 1 + 3.44T + 97T^{2} \) |
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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
−7.79343727197671239659557192291, −7.53153815093145887131648605072, −6.80444615704222569967655356485, −5.51213660530358592644735894807, −4.96134236081494230631000581196, −4.01161820107960298512040286371, −3.37454436583719963900220264247, −2.72480664104374685383781213402, −1.85019269731572047566416274905, 0,
1.85019269731572047566416274905, 2.72480664104374685383781213402, 3.37454436583719963900220264247, 4.01161820107960298512040286371, 4.96134236081494230631000581196, 5.51213660530358592644735894807, 6.80444615704222569967655356485, 7.53153815093145887131648605072, 7.79343727197671239659557192291