| L(s) = 1 | + (−1.50 + 0.198i)2-s + (−1.09 − 2.22i)3-s + (0.299 − 0.0801i)4-s + (−0.751 + 0.659i)5-s + (2.09 + 3.13i)6-s + (0.556 − 2.58i)7-s + (2.37 − 0.983i)8-s + (−1.92 + 2.51i)9-s + (1.00 − 1.14i)10-s + (1.65 − 0.108i)11-s + (−0.507 − 0.578i)12-s + (4.88 + 4.88i)13-s + (−0.324 + 4.00i)14-s + (2.29 + 0.950i)15-s + (−3.91 + 2.26i)16-s + (2.16 − 3.50i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 + 0.140i)2-s + (−0.634 − 1.28i)3-s + (0.149 − 0.0400i)4-s + (−0.336 + 0.294i)5-s + (0.856 + 1.28i)6-s + (0.210 − 0.977i)7-s + (0.839 − 0.347i)8-s + (−0.642 + 0.837i)9-s + (0.316 − 0.361i)10-s + (0.499 − 0.0327i)11-s + (−0.146 − 0.166i)12-s + (1.35 + 1.35i)13-s + (−0.0868 + 1.07i)14-s + (0.592 + 0.245i)15-s + (−0.979 + 0.565i)16-s + (0.525 − 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.370930 - 0.497942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.370930 - 0.497942i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.751 - 0.659i)T \) |
| 7 | \( 1 + (-0.556 + 2.58i)T \) |
| 17 | \( 1 + (-2.16 + 3.50i)T \) |
| good | 2 | \( 1 + (1.50 - 0.198i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (1.09 + 2.22i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (-1.65 + 0.108i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-4.88 - 4.88i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.736 + 5.59i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-8.18 - 4.03i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.401 - 2.01i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (0.208 - 0.102i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.192 + 2.93i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.543 - 2.73i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (4.53 + 10.9i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 4.60i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.794 - 1.03i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (6.26 + 0.824i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (0.207 + 0.611i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-2.46 - 1.42i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.40 - 2.94i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (10.5 + 3.57i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-3.01 + 6.11i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-2.93 + 7.08i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-3.31 - 0.888i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (2.62 - 0.522i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.57251545515017215428447103121, −9.272202789821074596805943113856, −8.693788572425604973201909755522, −7.44697606528955987391365743856, −7.09795909039981041579452106544, −6.49470638079234899397737802516, −4.89160536896830648443342075259, −3.67011845306170811566810039499, −1.56933421229308237118139908553, −0.70134166295548486804259279620,
1.23778590572487255955573888807, 3.33974370193615530659484581272, 4.46085163415719626395067497574, 5.38685075197236989477320489930, 6.17424563756011951195330142990, 7.993944946603199234776945010515, 8.468031505186552390187443643228, 9.256506229534434722016173191486, 10.07809800096914392291180787858, 10.77753051426906385447222912253
