L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 5-s − 1.61·6-s + 0.618·7-s − 8-s + 1.61·9-s + 10-s + 1.61·11-s + 1.61·12-s − 0.618·13-s − 0.618·14-s − 1.61·15-s + 16-s − 1.61·17-s − 1.61·18-s − 0.618·19-s − 20-s + 1.00·21-s − 1.61·22-s + 23-s − 1.61·24-s + 25-s + 0.618·26-s + 27-s + 0.618·28-s + ⋯ |
L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 5-s − 1.61·6-s + 0.618·7-s − 8-s + 1.61·9-s + 10-s + 1.61·11-s + 1.61·12-s − 0.618·13-s − 0.618·14-s − 1.61·15-s + 16-s − 1.61·17-s − 1.61·18-s − 0.618·19-s − 20-s + 1.00·21-s − 1.61·22-s + 23-s − 1.61·24-s + 25-s + 0.618·26-s + 27-s + 0.618·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.035712878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035712878\) |
\(L(1)\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 1.61T + T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - 1.61T + T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 + 1.61T + T^{2} \) |
| 19 | \( 1 + 0.618T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.980363312487646504683161782652, −9.061796910629116234925801099776, −8.662500869620582249474111113838, −8.113302862645078986967588394165, −7.08377169034451604008318923066, −6.70073983203313345622667871542, −4.61269364281420865638447359948, −3.74092091970217679947905169526, −2.68299827962395641366076845175, −1.59503123749376949809248399495,
1.59503123749376949809248399495, 2.68299827962395641366076845175, 3.74092091970217679947905169526, 4.61269364281420865638447359948, 6.70073983203313345622667871542, 7.08377169034451604008318923066, 8.113302862645078986967588394165, 8.662500869620582249474111113838, 9.061796910629116234925801099776, 9.980363312487646504683161782652