| L(s) = 1 | + (−0.0594 − 0.0745i)2-s + (−0.863 − 2.20i)3-s + (0.443 − 1.94i)4-s + (−0.284 + 3.80i)5-s + (−0.112 + 0.195i)6-s + (1.30 + 2.26i)7-s + (−0.342 + 0.165i)8-s + (−1.89 + 1.76i)9-s + (0.300 − 0.204i)10-s + (−0.456 − 1.99i)11-s + (−4.65 + 0.701i)12-s + (−1.76 − 1.20i)13-s + (0.0912 − 0.232i)14-s + (8.61 − 2.65i)15-s + (−3.55 − 1.71i)16-s + (0.388 + 5.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.0420 − 0.0527i)2-s + (−0.498 − 1.27i)3-s + (0.221 − 0.970i)4-s + (−0.127 + 1.70i)5-s + (−0.0460 + 0.0797i)6-s + (0.495 + 0.857i)7-s + (−0.121 + 0.0583i)8-s + (−0.632 + 0.586i)9-s + (0.0950 − 0.0647i)10-s + (−0.137 − 0.602i)11-s + (−1.34 + 0.202i)12-s + (−0.490 − 0.334i)13-s + (0.0243 − 0.0621i)14-s + (2.22 − 0.685i)15-s + (−0.888 − 0.427i)16-s + (0.0942 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.654776 - 0.278153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.654776 - 0.278153i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.37 - 5.62i)T \) |
| good | 2 | \( 1 + (0.0594 + 0.0745i)T + (-0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.863 + 2.20i)T + (-2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (0.284 - 3.80i)T + (-4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 2.26i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.456 + 1.99i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (1.76 + 1.20i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.388 - 5.18i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (0.603 + 0.559i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (2.60 + 0.805i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.43 + 3.65i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-7.40 + 1.11i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (0.431 - 0.746i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.40 + 3.02i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (0.00195 - 0.00857i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-3.12 + 2.13i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (3.54 + 1.70i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-6.20 - 0.935i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (9.35 + 8.67i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (0.220 - 0.0680i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-9.06 - 6.18i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.133 + 0.231i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.09 + 7.88i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.18 - 13.2i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.14 - 5.00i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−15.44018945550033736417069962145, −14.74292975502224551252131695484, −13.69536173642192823166911606557, −12.07431429274797071209897750462, −11.19314816300291583673725707291, −10.19809833641236158842366693776, −7.984872399903339848426424396279, −6.58364243771858094350978470580, −5.87448090368001811104597708925, −2.32730135834479592384162134235,
4.23832659534700481594411769385, 4.93512124014629427529553244753, 7.51357447377222093370703895554, 8.858419688532601907530838132622, 10.01899718333633113515756315939, 11.55329088961036395722754396973, 12.40230650066543945243423717096, 13.74590250941086358159097392554, 15.57642331026612472860241612133, 16.34220440689439118116022325864
