L(s) = 1 | − 1.61·2-s + 1.61·4-s − 8-s + 9-s + 0.618·11-s + 0.618·13-s − 1.61·17-s − 1.61·18-s − 1.61·19-s − 1.00·22-s + 25-s − 1.00·26-s − 1.61·31-s + 32-s + 2.61·34-s + 1.61·36-s − 1.61·37-s + 2.61·38-s + 0.618·41-s + 1.00·44-s + 0.618·47-s + 49-s − 1.61·50-s + 1.00·52-s − 1.61·61-s + 2.61·62-s − 1.61·64-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 1.61·4-s − 8-s + 9-s + 0.618·11-s + 0.618·13-s − 1.61·17-s − 1.61·18-s − 1.61·19-s − 1.00·22-s + 25-s − 1.00·26-s − 1.61·31-s + 32-s + 2.61·34-s + 1.61·36-s − 1.61·37-s + 2.61·38-s + 0.618·41-s + 1.00·44-s + 0.618·47-s + 49-s − 1.61·50-s + 1.00·52-s − 1.61·61-s + 2.61·62-s − 1.61·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3276305465\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3276305465\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 127 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.618T + T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 + 1.61T + T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.618T + T^{2} \) |
| 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 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−13.47280047519811087717932057367, −12.42445559467085815708960642182, −10.94914769092000989049753263230, −10.55681981503439842271293523294, −9.134503101340332623126892173935, −8.692856212946880856357580576314, −7.24533615331990691556672324869, −6.48490417853134628055577876098, −4.26406518992861025057344194770, −1.85920683345632168750464233606,
1.85920683345632168750464233606, 4.26406518992861025057344194770, 6.48490417853134628055577876098, 7.24533615331990691556672324869, 8.692856212946880856357580576314, 9.134503101340332623126892173935, 10.55681981503439842271293523294, 10.94914769092000989049753263230, 12.42445559467085815708960642182, 13.47280047519811087717932057367