| L(s) = 1 | − 2-s − 2.21·3-s + 4-s − 2.95·5-s + 2.21·6-s − 3.36·7-s − 8-s + 1.88·9-s + 2.95·10-s − 2.21·12-s − 3.45·13-s + 3.36·14-s + 6.53·15-s + 16-s + 17-s − 1.88·18-s + 0.923·19-s − 2.95·20-s + 7.44·21-s + 1.70·23-s + 2.21·24-s + 3.73·25-s + 3.45·26-s + 2.46·27-s − 3.36·28-s − 8.92·29-s − 6.53·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.27·3-s + 0.5·4-s − 1.32·5-s + 0.902·6-s − 1.27·7-s − 0.353·8-s + 0.628·9-s + 0.934·10-s − 0.638·12-s − 0.957·13-s + 0.899·14-s + 1.68·15-s + 0.250·16-s + 0.242·17-s − 0.444·18-s + 0.211·19-s − 0.660·20-s + 1.62·21-s + 0.355·23-s + 0.451·24-s + 0.746·25-s + 0.676·26-s + 0.473·27-s − 0.636·28-s − 1.65·29-s − 1.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 13 | \( 1 + 3.45T + 13T^{2} \) |
| 19 | \( 1 - 0.923T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 6.50T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 8.16T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 0.707T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 - 8.42T + 89T^{2} \) |
| 97 | \( 1 - 2.47T + 97T^{2} \) |
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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
−7.82189558348099668793960733133, −7.22864950221434245049164132029, −6.87791372969958724075190842107, −5.78524017758252531738211113495, −5.38414366616140253619366714136, −4.14154602032497433385779933685, −3.50495844318051750154535876884, −2.41470015484913153350356227364, −0.72098007057494037086739967075, 0,
0.72098007057494037086739967075, 2.41470015484913153350356227364, 3.50495844318051750154535876884, 4.14154602032497433385779933685, 5.38414366616140253619366714136, 5.78524017758252531738211113495, 6.87791372969958724075190842107, 7.22864950221434245049164132029, 7.82189558348099668793960733133
