L(s) = 1 | − 1.61·2-s + 0.618·3-s + 1.61·4-s − 1.00·6-s − 1.61·7-s − 8-s − 0.618·9-s + 1.00·12-s + 2.61·14-s + 0.618·17-s + 0.999·18-s − 1.00·21-s − 0.618·24-s + 25-s − 27-s − 2.61·28-s + 32-s − 1.00·34-s − 0.999·36-s + 0.618·37-s + 1.61·42-s + 47-s + 1.61·49-s − 1.61·50-s + 0.381·51-s − 1.61·53-s + 1.61·54-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 0.618·3-s + 1.61·4-s − 1.00·6-s − 1.61·7-s − 8-s − 0.618·9-s + 1.00·12-s + 2.61·14-s + 0.618·17-s + 0.999·18-s − 1.00·21-s − 0.618·24-s + 25-s − 27-s − 2.61·28-s + 32-s − 1.00·34-s − 0.999·36-s + 0.618·37-s + 1.61·42-s + 47-s + 1.61·49-s − 1.61·50-s + 0.381·51-s − 1.61·53-s + 1.61·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2477426425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2477426425\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 - 0.618T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 + 1.61T + T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 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
−16.39761299926307357992794402644, −15.30703876774217792567430703967, −13.81816516142158278342030488727, −12.42821254608935038877218490071, −10.84649755632872285874832993509, −9.667695505824941703172707285159, −8.974116803878374901191307120128, −7.70850801210328648392120929641, −6.34108978922784854587300086860, −2.98743469578434504405692118290,
2.98743469578434504405692118290, 6.34108978922784854587300086860, 7.70850801210328648392120929641, 8.974116803878374901191307120128, 9.667695505824941703172707285159, 10.84649755632872285874832993509, 12.42821254608935038877218490071, 13.81816516142158278342030488727, 15.30703876774217792567430703967, 16.39761299926307357992794402644