L(s) = 1 | + 1.27·2-s − 2.57·3-s − 0.370·4-s − 4.09·5-s − 3.28·6-s + 7-s − 3.02·8-s + 3.63·9-s − 5.22·10-s + 0.954·12-s − 4.39·13-s + 1.27·14-s + 10.5·15-s − 3.12·16-s + 4.19·17-s + 4.64·18-s − 1.24·19-s + 1.51·20-s − 2.57·21-s + 4.97·23-s + 7.79·24-s + 11.7·25-s − 5.60·26-s − 1.63·27-s − 0.370·28-s − 1.93·29-s + 13.4·30-s + ⋯ |
L(s) = 1 | + 0.902·2-s − 1.48·3-s − 0.185·4-s − 1.82·5-s − 1.34·6-s + 0.377·7-s − 1.06·8-s + 1.21·9-s − 1.65·10-s + 0.275·12-s − 1.21·13-s + 0.341·14-s + 2.72·15-s − 0.780·16-s + 1.01·17-s + 1.09·18-s − 0.286·19-s + 0.338·20-s − 0.562·21-s + 1.03·23-s + 1.59·24-s + 2.34·25-s − 1.09·26-s − 0.315·27-s − 0.0699·28-s − 0.359·29-s + 2.45·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5848104470\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5848104470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 3 | \( 1 + 2.57T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 0.716T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 3.97T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.59T + 67T^{2} \) |
| 71 | \( 1 - 0.218T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 4.56T + 79T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 - 4.20T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−10.60327954177841003438783236896, −9.453267789411852547784652967987, −8.266287448270104256214661159899, −7.43160236775152290755465211903, −6.62030011044934897555107744720, −5.39157546530575305744362327469, −4.89052175807114283420589181348, −4.16101749853542174053584074358, −3.14475765617848142011547352192, −0.56678963440996086866503791441,
0.56678963440996086866503791441, 3.14475765617848142011547352192, 4.16101749853542174053584074358, 4.89052175807114283420589181348, 5.39157546530575305744362327469, 6.62030011044934897555107744720, 7.43160236775152290755465211903, 8.266287448270104256214661159899, 9.453267789411852547784652967987, 10.60327954177841003438783236896