L(s) = 1 | + 0.614·2-s − 2.75·3-s − 1.62·4-s − 1.26·5-s − 1.69·6-s − 3.83·7-s − 2.22·8-s + 4.59·9-s − 0.775·10-s + 1.37·11-s + 4.47·12-s − 7.02·13-s − 2.35·14-s + 3.48·15-s + 1.87·16-s + 17-s + 2.82·18-s − 2.98·19-s + 2.04·20-s + 10.5·21-s + 0.845·22-s + 6.50·23-s + 6.13·24-s − 3.40·25-s − 4.31·26-s − 4.40·27-s + 6.22·28-s + ⋯ |
L(s) = 1 | + 0.434·2-s − 1.59·3-s − 0.811·4-s − 0.564·5-s − 0.691·6-s − 1.44·7-s − 0.786·8-s + 1.53·9-s − 0.245·10-s + 0.414·11-s + 1.29·12-s − 1.94·13-s − 0.629·14-s + 0.898·15-s + 0.469·16-s + 0.242·17-s + 0.665·18-s − 0.684·19-s + 0.458·20-s + 2.30·21-s + 0.180·22-s + 1.35·23-s + 1.25·24-s − 0.680·25-s − 0.846·26-s − 0.846·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2832137035\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2832137035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 0.614T + 2T^{2} \) |
| 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 + 1.26T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 - 9.00T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 47 | \( 1 - 0.246T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.613T + 97T^{2} \) |
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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
−10.30099177747652282005973641221, −9.770134934155492035981692073038, −8.918735390509469161690205864403, −7.42545025838795556541579272768, −6.67070673822455907984640408471, −5.83505947434126245224568845367, −4.95425340466721828557212592103, −4.23466067917262679587590092818, −3.03947327930515553483900731028, −0.42936050096662434865186241315,
0.42936050096662434865186241315, 3.03947327930515553483900731028, 4.23466067917262679587590092818, 4.95425340466721828557212592103, 5.83505947434126245224568845367, 6.67070673822455907984640408471, 7.42545025838795556541579272768, 8.918735390509469161690205864403, 9.770134934155492035981692073038, 10.30099177747652282005973641221