L(s) = 1 | − 1.73·2-s + 2·3-s + 0.999·4-s + 1.73·5-s − 3.46·6-s + 1.73·8-s + 9-s − 2.99·10-s + 1.99·12-s + 3.46·15-s − 5·16-s + 3·17-s − 1.73·18-s + 3.46·19-s + 1.73·20-s + 6·23-s + 3.46·24-s − 2.00·25-s − 4·27-s + 3·29-s − 5.99·30-s − 3.46·31-s + 5.19·32-s − 5.19·34-s + 0.999·36-s − 8.66·37-s − 5.99·38-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 1.15·3-s + 0.499·4-s + 0.774·5-s − 1.41·6-s + 0.612·8-s + 0.333·9-s − 0.948·10-s + 0.577·12-s + 0.894·15-s − 1.25·16-s + 0.727·17-s − 0.408·18-s + 0.794·19-s + 0.387·20-s + 1.25·23-s + 0.707·24-s − 0.400·25-s − 0.769·27-s + 0.557·29-s − 1.09·30-s − 0.622·31-s + 0.918·32-s − 0.891·34-s + 0.166·36-s − 1.42·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9663863038\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9663863038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−13.07149786603394019910645113250, −11.52515822423401509812583099945, −10.23881727093134391009984706507, −9.588508745175309869068694658006, −8.800662547309621166921013579389, −7.999777026071579701241342315621, −6.94378083482087772843213542582, −5.21823114907427806413664401822, −3.26403351690720069103565525424, −1.71909036867686537100924621887,
1.71909036867686537100924621887, 3.26403351690720069103565525424, 5.21823114907427806413664401822, 6.94378083482087772843213542582, 7.999777026071579701241342315621, 8.800662547309621166921013579389, 9.588508745175309869068694658006, 10.23881727093134391009984706507, 11.52515822423401509812583099945, 13.07149786603394019910645113250