| L(s) = 1 | + (−2.46 − 0.172i)2-s + (−0.0802 − 2.29i)3-s + (4.04 + 0.569i)4-s + (1.58 + 1.57i)5-s + (−0.198 + 5.67i)6-s + (−0.754 − 1.30i)7-s + (−5.04 − 1.07i)8-s + (−2.28 + 0.159i)9-s + (−3.62 − 4.15i)10-s + (0.143 + 1.36i)11-s + (0.983 − 9.35i)12-s + (−0.976 − 1.44i)13-s + (1.63 + 3.34i)14-s + (3.50 − 3.76i)15-s + (4.36 + 1.25i)16-s + (1.57 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−1.74 − 0.121i)2-s + (−0.0463 − 1.32i)3-s + (2.02 + 0.284i)4-s + (0.708 + 0.705i)5-s + (−0.0808 + 2.31i)6-s + (−0.285 − 0.493i)7-s + (−1.78 − 0.379i)8-s + (−0.761 + 0.0532i)9-s + (−1.14 − 1.31i)10-s + (0.0431 + 0.410i)11-s + (0.283 − 2.70i)12-s + (−0.270 − 0.401i)13-s + (0.436 + 0.894i)14-s + (0.903 − 0.973i)15-s + (1.09 + 0.313i)16-s + (0.381 − 0.945i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.372942 - 0.532829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372942 - 0.532829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.58 - 1.57i)T \) |
| 19 | \( 1 + (-3.54 + 2.54i)T \) |
| good | 2 | \( 1 + (2.46 + 0.172i)T + (1.98 + 0.278i)T^{2} \) |
| 3 | \( 1 + (0.0802 + 2.29i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (0.754 + 1.30i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.143 - 1.36i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (0.976 + 1.44i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 3.89i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-4.81 - 4.65i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.01 + 2.51i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (-4.17 + 4.63i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (7.68 + 5.58i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.04 + 0.874i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.767 + 4.35i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.790 + 1.95i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-4.18 - 0.588i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (-2.54 - 10.2i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (9.62 + 9.29i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-0.916 - 0.572i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (-0.506 - 0.269i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (4.39 - 6.51i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (0.212 + 6.08i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (-0.157 + 0.175i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-3.38 + 0.969i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (13.4 - 8.39i)T + (42.5 - 87.1i)T^{2} \) |
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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
−10.48422909123436734869633687581, −9.777926435332549735725927020328, −9.081827143876006618945495646723, −7.76966539189820547390494141626, −7.16352169278859046338537379536, −6.85276539582261770158934394563, −5.55116192547531211003488295323, −2.99488922203427132578588545248, −1.98125768825837825617711784752, −0.76135613208774109903582384506,
1.44152961061975023521385121286, 3.07548102651274775918310407707, 4.72440696681314053370434741793, 5.77255006829144965408364428550, 6.79126020316407290165049573487, 8.231491123334979519356222051629, 8.869119600109114900699827068917, 9.435500967771919606123762442517, 10.20236825717524501359566000915, 10.62222952869387063310048592239
