| L(s) = 1 | + (−0.800 + 1.16i)2-s + (1.21 − 1.02i)3-s + (−0.718 − 1.86i)4-s + (2.19 + 0.798i)5-s + (0.216 + 2.23i)6-s + (−1.56 − 0.905i)7-s + (2.75 + 0.655i)8-s + (−0.0835 + 0.474i)9-s + (−2.68 + 1.91i)10-s + (−1.01 + 0.586i)11-s + (−2.77 − 1.53i)12-s + (−2.13 + 2.54i)13-s + (2.31 − 1.10i)14-s + (3.48 − 1.26i)15-s + (−2.96 + 2.68i)16-s + (−1.18 − 6.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.565 + 0.824i)2-s + (0.701 − 0.588i)3-s + (−0.359 − 0.933i)4-s + (0.981 + 0.357i)5-s + (0.0883 + 0.912i)6-s + (−0.593 − 0.342i)7-s + (0.972 + 0.231i)8-s + (−0.0278 + 0.158i)9-s + (−0.850 + 0.607i)10-s + (−0.306 + 0.176i)11-s + (−0.801 − 0.443i)12-s + (−0.591 + 0.704i)13-s + (0.617 − 0.295i)14-s + (0.899 − 0.327i)15-s + (−0.741 + 0.670i)16-s + (−0.288 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871981 + 0.198922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871981 + 0.198922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.800 - 1.16i)T \) |
| 19 | \( 1 + (2.39 + 3.64i)T \) |
| good | 3 | \( 1 + (-1.21 + 1.02i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.19 - 0.798i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.56 + 0.905i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.01 - 0.586i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.13 - 2.54i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.18 + 6.74i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 3.32i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.974 + 0.171i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.07 - 8.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.26iT - 37T^{2} \) |
| 41 | \( 1 + (-6.74 - 8.03i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.85 + 7.83i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.49 - 1.14i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.859 + 2.36i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0105 + 0.0596i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.43 + 0.887i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.731 + 4.14i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-14.6 - 5.32i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-5.56 + 4.66i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (2.92 - 2.45i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.29 + 4.20i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.45 - 5.31i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.53 + 1.68i)T + (91.1 - 33.1i)T^{2} \) |
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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
−14.31449543733849385830699519946, −13.85878863280714755906988065323, −12.97409273860328384617255339812, −10.94849810752847678696932158810, −9.708311694149606723918270858114, −8.981967044246494341022224296573, −7.38551739844580352841927990059, −6.76943324381202753985908667710, −5.14116342442512256640866077235, −2.32820088326045019862488850381,
2.41875925243608617176042322461, 3.94113873222161072322377100891, 5.94581923075025526079999530819, 8.077303680579530211011557089163, 9.106561553163136820843184757342, 9.846310639404854818491804767634, 10.72003373992897548636582513133, 12.54060257851516927701613986333, 13.00002194303227768072499605990, 14.38511544397075865241159814025
