| L(s) = 1 | + 2-s + 4-s + 0.618·5-s + 8-s + 9-s + 0.618·10-s − 1.61·13-s + 16-s − 1.61·17-s + 18-s + 0.618·20-s − 0.618·25-s − 1.61·26-s − 1.61·29-s + 32-s − 1.61·34-s + 36-s + 0.618·37-s + 0.618·40-s + 0.618·41-s + 0.618·45-s + 49-s − 0.618·50-s − 1.61·52-s + 0.618·53-s − 1.61·58-s − 1.61·61-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 0.618·5-s + 8-s + 9-s + 0.618·10-s − 1.61·13-s + 16-s − 1.61·17-s + 18-s + 0.618·20-s − 0.618·25-s − 1.61·26-s − 1.61·29-s + 32-s − 1.61·34-s + 36-s + 0.618·37-s + 0.618·40-s + 0.618·41-s + 0.618·45-s + 49-s − 0.618·50-s − 1.61·52-s + 0.618·53-s − 1.61·58-s − 1.61·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.203786961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203786961\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.618T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−9.788309088208788376460864511642, −9.165280137550611520540442374426, −7.70259492117001425523138851049, −7.19130116723388732080911721109, −6.38357226037683613251325512203, −5.48533769426723532315148175920, −4.62732332062340080010821326124, −3.97400568413910300898318330206, −2.52927368835307880466437208044, −1.87611444413021101515239433871,
1.87611444413021101515239433871, 2.52927368835307880466437208044, 3.97400568413910300898318330206, 4.62732332062340080010821326124, 5.48533769426723532315148175920, 6.38357226037683613251325512203, 7.19130116723388732080911721109, 7.70259492117001425523138851049, 9.165280137550611520540442374426, 9.788309088208788376460864511642
