| L(s) = 1 | − 1.29·2-s − 6.31·4-s − 7.05·5-s − 33.4·7-s + 18.5·8-s + 9.15·10-s + 27.5·11-s − 55.6·13-s + 43.4·14-s + 26.3·16-s − 108.·17-s − 141.·19-s + 44.5·20-s − 35.8·22-s − 142.·23-s − 75.2·25-s + 72.2·26-s + 211.·28-s + 97.2·29-s − 221.·31-s − 182.·32-s + 140.·34-s + 235.·35-s + 339.·37-s + 183.·38-s − 131.·40-s + 266.·41-s + ⋯ |
| L(s) = 1 | − 0.458·2-s − 0.789·4-s − 0.631·5-s − 1.80·7-s + 0.821·8-s + 0.289·10-s + 0.756·11-s − 1.18·13-s + 0.828·14-s + 0.412·16-s − 1.54·17-s − 1.70·19-s + 0.498·20-s − 0.347·22-s − 1.29·23-s − 0.601·25-s + 0.545·26-s + 1.42·28-s + 0.622·29-s − 1.28·31-s − 1.01·32-s + 0.710·34-s + 1.13·35-s + 1.51·37-s + 0.784·38-s − 0.518·40-s + 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1202408228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1202408228\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 + 1.29T + 8T^{2} \) |
| 5 | \( 1 + 7.05T + 125T^{2} \) |
| 7 | \( 1 + 33.4T + 343T^{2} \) |
| 11 | \( 1 - 27.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 97.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 221.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 380.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 15.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 172.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 616.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 210.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 543.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 350.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 625.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15790876853716518082508651279, −9.470799113312530522177385618303, −8.891013154925630555154738290621, −7.84005725680249105041544396244, −6.82484221462284757157683997007, −6.03725043602580055236757133596, −4.36337500532918387710109432083, −3.91137524599700695985759427853, −2.35668290582042680506277613068, −0.21396072412968705187311391049,
0.21396072412968705187311391049, 2.35668290582042680506277613068, 3.91137524599700695985759427853, 4.36337500532918387710109432083, 6.03725043602580055236757133596, 6.82484221462284757157683997007, 7.84005725680249105041544396244, 8.891013154925630555154738290621, 9.470799113312530522177385618303, 10.15790876853716518082508651279
