| L(s) = 1 | − 2-s − 0.618·3-s + 4-s − 5-s + 0.618·6-s + 1.61·7-s − 8-s − 0.618·9-s + 10-s − 0.618·12-s − 0.618·13-s − 1.61·14-s + 0.618·15-s + 16-s + 1.61·17-s + 0.618·18-s − 20-s − 1.00·21-s − 0.618·23-s + 0.618·24-s + 25-s + 0.618·26-s + 27-s + 1.61·28-s + 29-s − 0.618·30-s − 1.61·31-s + ⋯ |
| L(s) = 1 | − 2-s − 0.618·3-s + 4-s − 5-s + 0.618·6-s + 1.61·7-s − 8-s − 0.618·9-s + 10-s − 0.618·12-s − 0.618·13-s − 1.61·14-s + 0.618·15-s + 16-s + 1.61·17-s + 0.618·18-s − 20-s − 1.00·21-s − 0.618·23-s + 0.618·24-s + 25-s + 0.618·26-s + 27-s + 1.61·28-s + 29-s − 0.618·30-s − 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5219206813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5219206813\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.618T + T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - 1.61T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.618T + T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.61T + T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.61T + T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−10.14776487787839366630307220676, −8.993980802057892798736855870154, −8.187341368449714849320501849874, −7.76319415504557478906703164933, −7.01376144202541873519439180911, −5.70082867672142893625880834378, −5.09280860801039267824164737614, −3.79802567714649067030651591111, −2.45839182994641168561114304917, −0.994557292542191562527591426035,
0.994557292542191562527591426035, 2.45839182994641168561114304917, 3.79802567714649067030651591111, 5.09280860801039267824164737614, 5.70082867672142893625880834378, 7.01376144202541873519439180911, 7.76319415504557478906703164933, 8.187341368449714849320501849874, 8.993980802057892798736855870154, 10.14776487787839366630307220676
