| L(s) = 1 | + 0.618·2-s − 0.618·4-s − 8-s + 9-s − 1.61·11-s − 1.61·13-s + 0.618·17-s + 0.618·18-s + 0.618·19-s − 1.00·22-s + 25-s − 1.00·26-s + 0.618·31-s + 0.999·32-s + 0.381·34-s − 0.618·36-s + 0.618·37-s + 0.381·38-s − 1.61·41-s + 0.999·44-s − 1.61·47-s + 49-s + 0.618·50-s + 0.999·52-s + 0.618·61-s + 0.381·62-s + 0.618·64-s + ⋯ |
| L(s) = 1 | + 0.618·2-s − 0.618·4-s − 8-s + 9-s − 1.61·11-s − 1.61·13-s + 0.618·17-s + 0.618·18-s + 0.618·19-s − 1.00·22-s + 25-s − 1.00·26-s + 0.618·31-s + 0.999·32-s + 0.381·34-s − 0.618·36-s + 0.618·37-s + 0.381·38-s − 1.61·41-s + 0.999·44-s − 1.61·47-s + 49-s + 0.618·50-s + 0.999·52-s + 0.618·61-s + 0.381·62-s + 0.618·64-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6772697748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6772697748\) |
| \(L(1)\) |
|
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 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.61T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 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
−13.42370588620668565955718275798, −12.77787074675390494649691571006, −11.96568531091756821059460171339, −10.26636958165125741525006942691, −9.697520639371775939113980261152, −8.173263553916733694561930400661, −7.11502962858507382676754181005, −5.35328852554235248733096561064, −4.64627465124050951229319361935, −2.93544618436142802891119222963,
2.93544618436142802891119222963, 4.64627465124050951229319361935, 5.35328852554235248733096561064, 7.11502962858507382676754181005, 8.173263553916733694561930400661, 9.697520639371775939113980261152, 10.26636958165125741525006942691, 11.96568531091756821059460171339, 12.77787074675390494649691571006, 13.42370588620668565955718275798
