L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 2·7-s − 8-s + 9-s − 3·10-s + 5·11-s + 12-s − 7·13-s + 2·14-s + 3·15-s + 16-s − 17-s − 18-s + 7·19-s + 3·20-s − 2·21-s − 5·22-s + 4·23-s − 24-s + 4·25-s + 7·26-s + 27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.50·11-s + 0.288·12-s − 1.94·13-s + 0.534·14-s + 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.60·19-s + 0.670·20-s − 0.436·21-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 4/5·25-s + 1.37·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.182357297\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182357297\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.60131264603828165617907250860, −11.61273048285036444962877668347, −10.05252486013069065936241851747, −9.538738513600828220010452429252, −9.061010670239073320902911505164, −7.36260827061863180158325670551, −6.62475750166554661041789766388, −5.25566528367566277200195898472, −3.21935054931024025505407094893, −1.82338902456101278198546006187,
1.82338902456101278198546006187, 3.21935054931024025505407094893, 5.25566528367566277200195898472, 6.62475750166554661041789766388, 7.36260827061863180158325670551, 9.061010670239073320902911505164, 9.538738513600828220010452429252, 10.05252486013069065936241851747, 11.61273048285036444962877668347, 12.60131264603828165617907250860