| L(s) = 1 | + 1.73·2-s − 0.732·3-s + 0.999·4-s + 5-s − 1.26·6-s + 0.732·7-s − 1.73·8-s − 2.46·9-s + 1.73·10-s + 1.26·11-s − 0.732·12-s − 4·13-s + 1.26·14-s − 0.732·15-s − 5·16-s − 17-s − 4.26·18-s + 5.46·19-s + 0.999·20-s − 0.535·21-s + 2.19·22-s + 2.19·23-s + 1.26·24-s + 25-s − 6.92·26-s + 4·27-s + 0.732·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.422·3-s + 0.499·4-s + 0.447·5-s − 0.517·6-s + 0.276·7-s − 0.612·8-s − 0.821·9-s + 0.547·10-s + 0.382·11-s − 0.211·12-s − 1.10·13-s + 0.338·14-s − 0.189·15-s − 1.25·16-s − 0.242·17-s − 1.00·18-s + 1.25·19-s + 0.223·20-s − 0.116·21-s + 0.468·22-s + 0.457·23-s + 0.258·24-s + 0.200·25-s − 1.35·26-s + 0.769·27-s + 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.429765515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429765515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−14.20204190271550574769746759430, −13.34314493158750539470958386483, −12.08362200800634378347322471186, −11.55816176180277487067958444215, −9.986651261973227704823650504951, −8.693059772496743559793207819411, −6.86478368489459895603477435819, −5.61469624409752576256081935994, −4.75821808947205352603115115491, −2.93032858041166400177262840994,
2.93032858041166400177262840994, 4.75821808947205352603115115491, 5.61469624409752576256081935994, 6.86478368489459895603477435819, 8.693059772496743559793207819411, 9.986651261973227704823650504951, 11.55816176180277487067958444215, 12.08362200800634378347322471186, 13.34314493158750539470958386483, 14.20204190271550574769746759430
