| L(s) = 1 | − 0.618·2-s − 0.618·4-s + 8-s + 9-s + 0.618·11-s − 0.618·13-s − 0.618·18-s − 1.61·19-s − 0.381·22-s + 1.61·23-s + 0.381·26-s + 0.618·31-s − 0.999·32-s − 0.618·36-s + 1.00·38-s − 0.381·44-s − 1.00·46-s + 49-s + 0.381·52-s − 0.381·62-s + 0.618·64-s + 1.61·67-s + 72-s + 1.61·73-s + 0.999·76-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.618·2-s − 0.618·4-s + 8-s + 9-s + 0.618·11-s − 0.618·13-s − 0.618·18-s − 1.61·19-s − 0.381·22-s + 1.61·23-s + 0.381·26-s + 0.618·31-s − 0.999·32-s − 0.618·36-s + 1.00·38-s − 0.381·44-s − 1.00·46-s + 49-s + 0.381·52-s − 0.381·62-s + 0.618·64-s + 1.61·67-s + 72-s + 1.61·73-s + 0.999·76-s + 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7783063178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7783063178\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.618T + T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.61T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 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
−9.270511619461808558546992796173, −8.749909487201758827769763861166, −7.909319299788355742157170532638, −7.09210009019073384321918082731, −6.46906154828270222925825075247, −5.13154463243713625867282813084, −4.48274635491993578970383196083, −3.72446257491871022433055357585, −2.22913284343448781901232224562, −1.02317678223516307668602072964,
1.02317678223516307668602072964, 2.22913284343448781901232224562, 3.72446257491871022433055357585, 4.48274635491993578970383196083, 5.13154463243713625867282813084, 6.46906154828270222925825075247, 7.09210009019073384321918082731, 7.909319299788355742157170532638, 8.749909487201758827769763861166, 9.270511619461808558546992796173
