| L(s) = 1 | + (−1.40 − 0.109i)2-s + (−1.68 − 0.383i)3-s + (1.97 + 0.308i)4-s + (1.56 − 1.03i)5-s + (2.33 + 0.724i)6-s + (−1.43 − 1.35i)7-s + (−2.75 − 0.651i)8-s + (2.70 + 1.29i)9-s + (−2.32 + 1.28i)10-s + (2.52 + 5.03i)11-s + (−3.21 − 1.27i)12-s + (2.28 − 0.985i)13-s + (1.87 + 2.06i)14-s + (−3.04 + 1.14i)15-s + (3.80 + 1.21i)16-s + (0.900 + 1.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0773i)2-s + (−0.975 − 0.221i)3-s + (0.988 + 0.154i)4-s + (0.701 − 0.461i)5-s + (0.955 + 0.295i)6-s + (−0.541 − 0.510i)7-s + (−0.973 − 0.230i)8-s + (0.902 + 0.431i)9-s + (−0.734 + 0.405i)10-s + (0.762 + 1.51i)11-s + (−0.929 − 0.368i)12-s + (0.633 − 0.273i)13-s + (0.500 + 0.551i)14-s + (−0.785 + 0.294i)15-s + (0.952 + 0.304i)16-s + (0.218 + 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.778977 - 0.129385i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778977 - 0.129385i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.109i)T \) |
| 3 | \( 1 + (1.68 + 0.383i)T \) |
| good | 5 | \( 1 + (-1.56 + 1.03i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (1.43 + 1.35i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.52 - 5.03i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (-2.28 + 0.985i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (-0.900 - 1.07i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (4.83 + 4.05i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-3.27 - 3.47i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.586 - 0.0685i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-0.791 - 3.33i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-6.70 + 1.18i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-6.63 - 4.94i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.361 + 6.21i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-10.9 - 2.60i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (5.68 + 9.84i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.55 - 3.09i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-13.2 + 3.97i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (2.81 + 0.329i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-2.45 - 0.894i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.31 + 2.66i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-12.1 + 9.07i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-6.07 + 4.52i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-2.62 - 7.20i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (7.99 + 5.25i)T + (38.4 + 89.0i)T^{2} \) |
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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
−10.47023509673326426691323257391, −9.608103295996578443818477462003, −9.139962889889403257345993366766, −7.76780945365073944030752924303, −6.86178877218771314475836052017, −6.36507394355129236867867659043, −5.23784555587660563824350567354, −3.96245083033518753620244866992, −2.07765753369418184685517895199, −0.983268316580396317107261677311,
0.939173555162395391047134050955, 2.55429225789283109756911492520, 3.92486680401274485479112370833, 5.87127049975464998208681643488, 6.05150025420097139684835396580, 6.75712275183689311953420893912, 8.150570101390909654060862311830, 9.105611465579630807307335264173, 9.682643077951805406988356104332, 10.77152612472573587562916139581
