| L(s) = 1 | − 2-s + 4-s + 0.618·5-s − 8-s + 9-s − 0.618·10-s + 1.61·13-s + 16-s − 1.61·17-s − 18-s + 0.618·20-s − 0.618·25-s − 1.61·26-s + 1.61·29-s − 32-s + 1.61·34-s + 36-s − 0.618·37-s − 0.618·40-s − 0.618·41-s + 0.618·45-s + 49-s + 0.618·50-s + 1.61·52-s − 0.618·53-s − 1.61·58-s − 1.61·61-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 0.618·5-s − 8-s + 9-s − 0.618·10-s + 1.61·13-s + 16-s − 1.61·17-s − 18-s + 0.618·20-s − 0.618·25-s − 1.61·26-s + 1.61·29-s − 32-s + 1.61·34-s + 36-s − 0.618·37-s − 0.618·40-s − 0.618·41-s + 0.618·45-s + 49-s + 0.618·50-s + 1.61·52-s − 0.618·53-s − 1.61·58-s − 1.61·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8828567398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8828567398\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.618T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618T + T^{2} \) |
| 41 | \( 1 + 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−9.689242347296838994365521450259, −8.878353664731854470088932679646, −8.383454299978431655208036264200, −7.31510960255196794872843257777, −6.49680650443331888950116910231, −6.04675445483723202641328081025, −4.67114994899042061909186054176, −3.55170664133589650237045826662, −2.23037934588408357862289591024, −1.31287046705749264617439250942,
1.31287046705749264617439250942, 2.23037934588408357862289591024, 3.55170664133589650237045826662, 4.67114994899042061909186054176, 6.04675445483723202641328081025, 6.49680650443331888950116910231, 7.31510960255196794872843257777, 8.383454299978431655208036264200, 8.878353664731854470088932679646, 9.689242347296838994365521450259
