| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 0.618·7-s + 8-s + 9-s − 1.61·11-s + 12-s + 4.85·13-s − 0.618·14-s + 16-s + 0.763·17-s + 18-s + 5.85·19-s − 0.618·21-s − 1.61·22-s + 4.85·23-s + 24-s + 4.85·26-s + 27-s − 0.618·28-s − 2.76·29-s − 2.47·31-s + 32-s − 1.61·33-s + 0.763·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.233·7-s + 0.353·8-s + 0.333·9-s − 0.487·11-s + 0.288·12-s + 1.34·13-s − 0.165·14-s + 0.250·16-s + 0.185·17-s + 0.235·18-s + 1.34·19-s − 0.134·21-s − 0.344·22-s + 1.01·23-s + 0.204·24-s + 0.951·26-s + 0.192·27-s − 0.116·28-s − 0.513·29-s − 0.444·31-s + 0.176·32-s − 0.281·33-s + 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.911128567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.911128567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 9.56T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−10.49901724246114201745245450906, −9.448312331585159139387796529332, −8.638418767219358869130203632791, −7.66130772095485932139952497465, −6.87193049468976660805782303143, −5.79376436374249688830188884938, −4.94335705396492460099575127121, −3.63383674900375100745391922346, −3.04912687077782528567868562596, −1.52018401436931715243075964872,
1.52018401436931715243075964872, 3.04912687077782528567868562596, 3.63383674900375100745391922346, 4.94335705396492460099575127121, 5.79376436374249688830188884938, 6.87193049468976660805782303143, 7.66130772095485932139952497465, 8.638418767219358869130203632791, 9.448312331585159139387796529332, 10.49901724246114201745245450906
