| L(s) = 1 | − 0.618·2-s + 0.618·3-s − 0.618·4-s − 0.381·6-s + 1.61·7-s + 8-s − 0.618·9-s − 11-s − 0.381·12-s − 13-s − 1.00·14-s + 0.381·18-s − 0.618·19-s + 1.00·21-s + 0.618·22-s − 1.61·23-s + 0.618·24-s + 25-s + 0.618·26-s − 27-s − 0.999·28-s − 0.999·32-s − 0.618·33-s + 0.381·36-s + 0.381·38-s − 0.618·39-s + 1.61·41-s + ⋯ |
| L(s) = 1 | − 0.618·2-s + 0.618·3-s − 0.618·4-s − 0.381·6-s + 1.61·7-s + 8-s − 0.618·9-s − 11-s − 0.381·12-s − 13-s − 1.00·14-s + 0.381·18-s − 0.618·19-s + 1.00·21-s + 0.618·22-s − 1.61·23-s + 0.618·24-s + 25-s + 0.618·26-s − 27-s − 0.999·28-s − 0.999·32-s − 0.618·33-s + 0.381·36-s + 0.381·38-s − 0.618·39-s + 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5167072467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5167072467\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + T^{2} \) |
| 3 | \( 1 - 0.618T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.618T + T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.61T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−13.60486632803815923946613303662, −12.36739753161912299402971421224, −11.09964703010420586370456732131, −10.21318557697500063287799539775, −9.004257973228336367496572581079, −8.129979544218070959974762466048, −7.66026602909536586139046396552, −5.40644236408523558557252902644, −4.38802451253424855324823205856, −2.27203371878102644767358401045,
2.27203371878102644767358401045, 4.38802451253424855324823205856, 5.40644236408523558557252902644, 7.66026602909536586139046396552, 8.129979544218070959974762466048, 9.004257973228336367496572581079, 10.21318557697500063287799539775, 11.09964703010420586370456732131, 12.36739753161912299402971421224, 13.60486632803815923946613303662
