L(s) = 1 | + 2-s + 1.30·3-s + 4-s − 4.16·5-s + 1.30·6-s − 4.89·7-s + 8-s − 1.28·9-s − 4.16·10-s − 5.16·11-s + 1.30·12-s − 4.89·14-s − 5.44·15-s + 16-s − 0.826·17-s − 1.28·18-s + 19-s − 4.16·20-s − 6.40·21-s − 5.16·22-s + 0.110·23-s + 1.30·24-s + 12.3·25-s − 5.61·27-s − 4.89·28-s − 7.43·29-s − 5.44·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.755·3-s + 0.5·4-s − 1.86·5-s + 0.534·6-s − 1.84·7-s + 0.353·8-s − 0.428·9-s − 1.31·10-s − 1.55·11-s + 0.377·12-s − 1.30·14-s − 1.40·15-s + 0.250·16-s − 0.200·17-s − 0.303·18-s + 0.229·19-s − 0.930·20-s − 1.39·21-s − 1.10·22-s + 0.0229·23-s + 0.267·24-s + 2.46·25-s − 1.07·27-s − 0.924·28-s − 1.38·29-s − 0.994·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6807365657\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6807365657\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 17 | \( 1 + 0.826T + 17T^{2} \) |
| 23 | \( 1 - 0.110T + 23T^{2} \) |
| 29 | \( 1 + 7.43T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 - 4.61T + 37T^{2} \) |
| 41 | \( 1 + 1.49T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 - 8.01T + 59T^{2} \) |
| 61 | \( 1 + 7.50T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 + 6.62T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−7.77376431159304828872131986049, −7.47185556772592987936472014755, −6.74828876959189449508126137381, −5.82715796808369581309682539915, −5.11997476815588167571278942512, −4.03618011957054031455028427389, −3.56744242348858741252344992095, −3.01049294199019768818326138664, −2.43945110011142949566804889235, −0.33577518031953326290975071511,
0.33577518031953326290975071511, 2.43945110011142949566804889235, 3.01049294199019768818326138664, 3.56744242348858741252344992095, 4.03618011957054031455028427389, 5.11997476815588167571278942512, 5.82715796808369581309682539915, 6.74828876959189449508126137381, 7.47185556772592987936472014755, 7.77376431159304828872131986049