| L(s) = 1 | + (0.694 + 2.36i)2-s + (−1.73 − 0.0101i)3-s + (−3.43 + 2.20i)4-s + (−0.0722 − 0.502i)5-s + (−1.17 − 4.10i)6-s + (3.47 + 1.58i)7-s + (−3.87 − 3.35i)8-s + (2.99 + 0.0352i)9-s + (1.13 − 0.519i)10-s + (0.971 + 0.285i)11-s + (5.96 − 3.78i)12-s + (−1.96 − 4.29i)13-s + (−1.34 + 9.32i)14-s + (0.120 + 0.871i)15-s + (1.85 − 4.07i)16-s + (−5.03 − 3.23i)17-s + ⋯ |
| L(s) = 1 | + (0.491 + 1.67i)2-s + (−0.999 − 0.00587i)3-s + (−1.71 + 1.10i)4-s + (−0.0323 − 0.224i)5-s + (−0.481 − 1.67i)6-s + (1.31 + 0.599i)7-s + (−1.36 − 1.18i)8-s + (0.999 + 0.0117i)9-s + (0.360 − 0.164i)10-s + (0.292 + 0.0859i)11-s + (1.72 − 1.09i)12-s + (−0.544 − 1.19i)13-s + (−0.358 + 2.49i)14-s + (0.0309 + 0.224i)15-s + (0.464 − 1.01i)16-s + (−1.22 − 0.784i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.384752 + 0.797615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384752 + 0.797615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 + 0.0101i)T \) |
| 23 | \( 1 + (-3.68 + 3.07i)T \) |
| good | 2 | \( 1 + (-0.694 - 2.36i)T + (-1.68 + 1.08i)T^{2} \) |
| 5 | \( 1 + (0.0722 + 0.502i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-3.47 - 1.58i)T + (4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.971 - 0.285i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.96 + 4.29i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.03 + 3.23i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.15 - 1.79i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.433 - 0.674i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.11 - 2.44i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (4.43 + 0.637i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.70 + 0.245i)T + (39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (4.77 - 4.13i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 0.781iT - 47T^{2} \) |
| 53 | \( 1 + (0.0144 - 0.0316i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.669 - 0.305i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (4.17 + 3.61i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (0.413 + 1.40i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-0.150 - 0.512i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.76 + 2.41i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (4.78 - 2.18i)T + (51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.40 + 9.76i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-11.1 - 12.8i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (4.01 - 0.577i)T + (93.0 - 27.3i)T^{2} \) |
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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
−15.21554953386078527041197499425, −14.47101641992710468816700708342, −13.10008958007055383178840155526, −12.13515084616202717483994373279, −10.84252259356676143613393779644, −8.947585196563157486339491661060, −7.78456252281168355150497665352, −6.65628833060548836566150550771, −5.26360556599998180486310692226, −4.75760594734291592780638524124,
1.68025888100039658857977930389, 4.15304081711855776393954260904, 5.01949495188115337124185161003, 7.01082969370801189337800224703, 9.127204406095053815379295047796, 10.51588266658897428588011756274, 11.22679166792034708976106270613, 11.75089633385916860781971081764, 12.99699157537804891410857853460, 13.97412834051651707735251206367
