| L(s) = 1 | + (1.51 − 0.0681i)2-s + (0.308 − 0.0277i)4-s + (−1.18 − 0.386i)5-s + (−0.789 − 2.86i)7-s + (−2.54 + 0.344i)8-s + (−1.83 − 0.505i)10-s + (−1.51 − 2.82i)11-s + (−0.301 − 0.162i)13-s + (−1.39 − 4.29i)14-s + (−4.45 + 0.807i)16-s + (0.322 − 0.234i)17-s + (2.72 − 6.37i)19-s + (−0.377 − 0.0862i)20-s + (−2.49 − 4.17i)22-s + (3.19 − 4.00i)23-s + ⋯ |
| L(s) = 1 | + (1.07 − 0.0482i)2-s + (0.154 − 0.0138i)4-s + (−0.531 − 0.172i)5-s + (−0.298 − 1.08i)7-s + (−0.899 + 0.121i)8-s + (−0.579 − 0.159i)10-s + (−0.457 − 0.850i)11-s + (−0.0836 − 0.0450i)13-s + (−0.372 − 1.14i)14-s + (−1.11 + 0.201i)16-s + (0.0781 − 0.0568i)17-s + (0.625 − 1.46i)19-s + (−0.0844 − 0.0192i)20-s + (−0.532 − 0.890i)22-s + (0.665 − 0.834i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.685754 - 1.17910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685754 - 1.17910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (-8.06 - 2.42i)T \) |
| good | 2 | \( 1 + (-1.51 + 0.0681i)T + (1.99 - 0.179i)T^{2} \) |
| 5 | \( 1 + (1.18 + 0.386i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (0.789 + 2.86i)T + (-6.00 + 3.59i)T^{2} \) |
| 11 | \( 1 + (1.51 + 2.82i)T + (-6.05 + 9.18i)T^{2} \) |
| 13 | \( 1 + (0.301 + 0.162i)T + (7.16 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.322 + 0.234i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.72 + 6.37i)T + (-13.1 - 13.7i)T^{2} \) |
| 23 | \( 1 + (-3.19 + 4.00i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (5.08 - 8.51i)T + (-13.7 - 25.5i)T^{2} \) |
| 31 | \( 1 + (0.817 - 4.50i)T + (-29.0 - 10.8i)T^{2} \) |
| 37 | \( 1 + (-0.537 - 0.673i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (4.66 - 2.24i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-11.8 + 4.45i)T + (32.3 - 28.2i)T^{2} \) |
| 47 | \( 1 + (0.789 + 0.689i)T + (6.30 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.21 - 0.108i)T + (52.1 + 9.46i)T^{2} \) |
| 59 | \( 1 + (6.57 - 6.88i)T + (-2.64 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-3.09 + 11.2i)T + (-52.3 - 31.2i)T^{2} \) |
| 67 | \( 1 + (-0.368 - 4.09i)T + (-65.9 + 11.9i)T^{2} \) |
| 73 | \( 1 + (0.573 + 12.7i)T + (-72.7 + 6.54i)T^{2} \) |
| 79 | \( 1 + (0.413 + 3.05i)T + (-76.1 + 21.0i)T^{2} \) |
| 83 | \( 1 + (1.12 + 1.07i)T + (3.72 + 82.9i)T^{2} \) |
| 89 | \( 1 + (0.464 - 5.16i)T + (-87.5 - 15.8i)T^{2} \) |
| 97 | \( 1 + (0.680 - 1.41i)T + (-60.4 - 75.8i)T^{2} \) |
| 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
−10.63535896556244974166515002439, −9.329911528341086982489938738434, −8.584350188647447621382728411505, −7.42801669698540878660129237512, −6.65071784986612345720435750759, −5.42971990175434666775467976799, −4.66810094665059265890895505685, −3.68876987648403648537368970291, −2.91797826707994433814291607415, −0.51241330432051423817597439179,
2.28583881507241040899003788686, 3.44812810331074433035843218775, 4.29731371360341650628456174519, 5.53393043047142347237189931595, 5.89347586089486349051331457558, 7.29137995049674000187581427858, 8.076575317451311060437854966539, 9.361691496814890626270447245386, 9.766732374352317054312045979569, 11.24794282031842096097294075108
