| L(s) = 1 | − 2-s + 4-s + 2.85·5-s + 1.23·7-s − 8-s − 2.85·10-s + 3.61·11-s + 3.85·13-s − 1.23·14-s + 16-s − 4.47·17-s − 4.47·19-s + 2.85·20-s − 3.61·22-s + 3.85·23-s + 3.14·25-s − 3.85·26-s + 1.23·28-s − 6.32·29-s + 9.61·31-s − 32-s + 4.47·34-s + 3.52·35-s − 37-s + 4.47·38-s − 2.85·40-s − 7.38·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.27·5-s + 0.467·7-s − 0.353·8-s − 0.902·10-s + 1.09·11-s + 1.06·13-s − 0.330·14-s + 0.250·16-s − 1.08·17-s − 1.02·19-s + 0.638·20-s − 0.771·22-s + 0.803·23-s + 0.629·25-s − 0.755·26-s + 0.233·28-s − 1.17·29-s + 1.72·31-s − 0.176·32-s + 0.766·34-s + 0.596·35-s − 0.164·37-s + 0.725·38-s − 0.451·40-s − 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.514588379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.514588379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−10.46362769212668555174233177355, −9.535346551387154414694290683943, −8.867376536083673390760119910492, −8.234962024644255586792061168563, −6.66905150130503432360910126425, −6.41681475910582889607095725592, −5.21403278492601116080479391320, −3.89633259338865014135668912905, −2.29679447527135897016348658064, −1.35434847827472147039815903809,
1.35434847827472147039815903809, 2.29679447527135897016348658064, 3.89633259338865014135668912905, 5.21403278492601116080479391320, 6.41681475910582889607095725592, 6.66905150130503432360910126425, 8.234962024644255586792061168563, 8.867376536083673390760119910492, 9.535346551387154414694290683943, 10.46362769212668555174233177355
