| L(s) = 1 | + 0.618·2-s − 0.618·4-s + 1.61·7-s − 8-s + 11-s − 13-s + 1.00·14-s − 0.618·19-s + 0.618·22-s + 1.61·23-s + 25-s − 0.618·26-s − 0.999·28-s + 0.999·32-s − 0.381·38-s − 1.61·41-s − 0.618·44-s + 1.00·46-s + 1.61·49-s + 0.618·50-s + 0.618·52-s − 0.618·53-s − 1.61·56-s + 0.618·64-s − 0.618·73-s + 0.381·76-s + 1.61·77-s + ⋯ |
| L(s) = 1 | + 0.618·2-s − 0.618·4-s + 1.61·7-s − 8-s + 11-s − 13-s + 1.00·14-s − 0.618·19-s + 0.618·22-s + 1.61·23-s + 25-s − 0.618·26-s − 0.999·28-s + 0.999·32-s − 0.381·38-s − 1.61·41-s − 0.618·44-s + 1.00·46-s + 1.61·49-s + 0.618·50-s + 0.618·52-s − 0.618·53-s − 1.61·56-s + 0.618·64-s − 0.618·73-s + 0.381·76-s + 1.61·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.418510106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418510106\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + T^{2} \) |
| 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
−9.778102263307145683477115407019, −8.858629888446140693401507279937, −8.461176844455255029107338654918, −7.36545295238542606497065935798, −6.51695104166117993467808438448, −5.23513571633783344978278439501, −4.85016054757954122068774475305, −4.06583679467251771159061010269, −2.84332762280610811392257984819, −1.41393594341147264303934065962,
1.41393594341147264303934065962, 2.84332762280610811392257984819, 4.06583679467251771159061010269, 4.85016054757954122068774475305, 5.23513571633783344978278439501, 6.51695104166117993467808438448, 7.36545295238542606497065935798, 8.461176844455255029107338654918, 8.858629888446140693401507279937, 9.778102263307145683477115407019
