| L(s) = 1 | + 0.239·2-s − 1.94·4-s + 1.76·5-s + 3.18·7-s − 0.942·8-s + 0.421·10-s − 0.181·11-s − 3.18·13-s + 0.760·14-s + 3.66·16-s + 7.84·17-s − 19-s − 3.42·20-s − 0.0435·22-s + 3.28·23-s − 1.89·25-s − 0.760·26-s − 6.18·28-s + 9.48·29-s + 2.18·31-s + 2.76·32-s + 1.87·34-s + 5.60·35-s − 6.42·37-s − 0.239·38-s − 1.66·40-s + 1.76·41-s + ⋯ |
| L(s) = 1 | + 0.169·2-s − 0.971·4-s + 0.787·5-s + 1.20·7-s − 0.333·8-s + 0.133·10-s − 0.0548·11-s − 0.882·13-s + 0.203·14-s + 0.915·16-s + 1.90·17-s − 0.229·19-s − 0.764·20-s − 0.00927·22-s + 0.684·23-s − 0.379·25-s − 0.149·26-s − 1.16·28-s + 1.76·29-s + 0.391·31-s + 0.488·32-s + 0.321·34-s + 0.947·35-s − 1.05·37-s − 0.0387·38-s − 0.262·40-s + 0.275·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.618074954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618074954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 0.181T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 - 7.84T + 17T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 - 2.18T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 8.78T + 59T^{2} \) |
| 61 | \( 1 + 0.703T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 3.22T + 71T^{2} \) |
| 73 | \( 1 + 6.88T + 73T^{2} \) |
| 79 | \( 1 + 0.986T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 7.58T + 89T^{2} \) |
| 97 | \( 1 - 0.784T + 97T^{2} \) |
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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
−10.67653457573925394040018015974, −9.980048169971945287842509918463, −9.189877923633491464300340158677, −8.216151758667408746486602932321, −7.49039901017856803908247726514, −5.94736461229081816354651280295, −5.16674489762225930925511890893, −4.44042427393843273203792995946, −2.92186123338344332732535032113, −1.30362266295686547122763651697,
1.30362266295686547122763651697, 2.92186123338344332732535032113, 4.44042427393843273203792995946, 5.16674489762225930925511890893, 5.94736461229081816354651280295, 7.49039901017856803908247726514, 8.216151758667408746486602932321, 9.189877923633491464300340158677, 9.980048169971945287842509918463, 10.67653457573925394040018015974
