L(s) = 1 | + 0.618·2-s − 0.618·4-s − 1.61·5-s − 8-s + 9-s − 1.00·10-s + 0.618·11-s + 0.618·13-s + 0.618·18-s − 1.61·19-s + 0.999·20-s + 0.381·22-s − 1.61·23-s + 1.61·25-s + 0.381·26-s + 0.618·31-s + 0.999·32-s − 0.618·36-s − 1.00·38-s + 1.61·40-s − 0.381·44-s − 1.61·45-s − 1.00·46-s + 49-s + 1.00·50-s − 0.381·52-s − 1.00·55-s + ⋯ |
L(s) = 1 | + 0.618·2-s − 0.618·4-s − 1.61·5-s − 8-s + 9-s − 1.00·10-s + 0.618·11-s + 0.618·13-s + 0.618·18-s − 1.61·19-s + 0.999·20-s + 0.381·22-s − 1.61·23-s + 1.61·25-s + 0.381·26-s + 0.618·31-s + 0.999·32-s − 0.618·36-s − 1.00·38-s + 1.61·40-s − 0.381·44-s − 1.61·45-s − 1.00·46-s + 49-s + 1.00·50-s − 0.381·52-s − 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5273458988\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5273458988\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 - T \) |
good | 2 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.61T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.618T + T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−14.79301903017583787602292860928, −13.55274723202347412692925946328, −12.47946133071320119957371003723, −11.81774686271880974772720205144, −10.41921929706756148110282082287, −8.873978905013962342505313744455, −7.87413839665017329527808259336, −6.38007356073625378647747108423, −4.39037048839329186211077735661, −3.84845369197390074372926468934,
3.84845369197390074372926468934, 4.39037048839329186211077735661, 6.38007356073625378647747108423, 7.87413839665017329527808259336, 8.873978905013962342505313744455, 10.41921929706756148110282082287, 11.81774686271880974772720205144, 12.47946133071320119957371003723, 13.55274723202347412692925946328, 14.79301903017583787602292860928