| L(s) = 1 | + 0.618·2-s − 0.618·4-s + 0.618·5-s + 0.618·7-s − 8-s + 9-s + 0.381·10-s − 1.61·11-s + 2·13-s + 0.381·14-s + 0.618·18-s − 1.61·19-s − 0.381·20-s − 1.00·22-s − 0.618·25-s + 1.23·26-s − 0.381·28-s − 1.61·29-s + 0.999·32-s + 0.381·35-s − 0.618·36-s − 1.00·38-s − 0.618·40-s + 0.999·44-s + 0.618·45-s − 0.618·49-s − 0.381·50-s + ⋯ |
| L(s) = 1 | + 0.618·2-s − 0.618·4-s + 0.618·5-s + 0.618·7-s − 8-s + 9-s + 0.381·10-s − 1.61·11-s + 2·13-s + 0.381·14-s + 0.618·18-s − 1.61·19-s − 0.381·20-s − 1.00·22-s − 0.618·25-s + 1.23·26-s − 0.381·28-s − 1.61·29-s + 0.999·32-s + 0.381·35-s − 0.618·36-s − 1.00·38-s − 0.618·40-s + 0.999·44-s + 0.618·45-s − 0.618·49-s − 0.381·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 439 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 439 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.099139040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099139040\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 439 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.618T + T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 - 2T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.618T + T^{2} \) |
| 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
−11.20263903430736668543307381806, −10.55504746645258310583271661051, −9.592559240148356599691103067688, −8.563270288510592776051511691231, −7.85158854435657084102546298222, −6.33273967941237368096244384005, −5.55982995978664929702525851156, −4.56971265245924120561328325259, −3.61226720279512701821535448768, −1.91292027741160208225810263818,
1.91292027741160208225810263818, 3.61226720279512701821535448768, 4.56971265245924120561328325259, 5.55982995978664929702525851156, 6.33273967941237368096244384005, 7.85158854435657084102546298222, 8.563270288510592776051511691231, 9.592559240148356599691103067688, 10.55504746645258310583271661051, 11.20263903430736668543307381806
