L(s) = 1 | + 0.618·2-s − 0.618·4-s + 7-s − 8-s + 9-s + 0.618·14-s + 0.618·18-s + 0.618·23-s + 25-s − 0.618·28-s − 1.61·29-s + 0.999·32-s − 0.618·36-s + 0.618·37-s − 1.61·43-s + 0.381·46-s + 49-s + 0.618·50-s − 1.61·53-s − 56-s − 1.00·58-s + 63-s + 0.618·64-s − 1.61·67-s − 1.61·71-s − 72-s + 0.381·74-s + ⋯ |
L(s) = 1 | + 0.618·2-s − 0.618·4-s + 7-s − 8-s + 9-s + 0.618·14-s + 0.618·18-s + 0.618·23-s + 25-s − 0.618·28-s − 1.61·29-s + 0.999·32-s − 0.618·36-s + 0.618·37-s − 1.61·43-s + 0.381·46-s + 49-s + 0.618·50-s − 1.61·53-s − 56-s − 1.00·58-s + 63-s + 0.618·64-s − 1.61·67-s − 1.61·71-s − 72-s + 0.381·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.270468421\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270468421\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 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
−10.42964538719874638291906914025, −9.488119784404378658679765650635, −8.764197609015200463055229292853, −7.83600409825000688713498094589, −6.96283083950172018601862587440, −5.78218156187590736851867784068, −4.86154637780817777370635776941, −4.31532800776367281280497504305, −3.17195155876334466872557958739, −1.54409029844824918147557079032,
1.54409029844824918147557079032, 3.17195155876334466872557958739, 4.31532800776367281280497504305, 4.86154637780817777370635776941, 5.78218156187590736851867784068, 6.96283083950172018601862587440, 7.83600409825000688713498094589, 8.764197609015200463055229292853, 9.488119784404378658679765650635, 10.42964538719874638291906914025