| L(s) = 1 | + (−1.07 − 0.915i)2-s + (0.755 − 0.654i)3-s + (0.322 + 1.97i)4-s + (2.07 − 0.298i)5-s + (−1.41 + 0.0136i)6-s + (1.09 − 2.39i)7-s + (1.45 − 2.42i)8-s + (0.142 − 0.989i)9-s + (−2.51 − 1.57i)10-s + (0.479 − 1.63i)11-s + (1.53 + 1.28i)12-s + (−2.58 + 1.17i)13-s + (−3.37 + 1.58i)14-s + (1.37 − 1.58i)15-s + (−3.79 + 1.27i)16-s + (2.37 + 1.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.762 − 0.647i)2-s + (0.436 − 0.378i)3-s + (0.161 + 0.986i)4-s + (0.928 − 0.133i)5-s + (−0.577 + 0.00556i)6-s + (0.414 − 0.907i)7-s + (0.516 − 0.856i)8-s + (0.0474 − 0.329i)9-s + (−0.793 − 0.499i)10-s + (0.144 − 0.492i)11-s + (0.443 + 0.369i)12-s + (−0.716 + 0.327i)13-s + (−0.903 + 0.422i)14-s + (0.354 − 0.409i)15-s + (−0.947 + 0.318i)16-s + (0.574 + 0.369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.947794 - 1.01586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.947794 - 1.01586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 + 0.915i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (-4.78 - 0.359i)T \) |
| good | 5 | \( 1 + (-2.07 + 0.298i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 2.39i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.479 + 1.63i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.58 - 1.17i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 1.52i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.187 + 0.291i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.23 + 3.47i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.65 + 4.22i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (7.11 + 1.02i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.262 - 1.82i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.53 - 2.19i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + (2.30 + 1.05i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (5.62 - 2.56i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (5.14 + 4.45i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.83 - 6.25i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-14.6 + 4.30i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (5.51 - 3.54i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.00 + 10.9i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (3.63 + 0.522i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.14 - 7.08i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.676 - 4.70i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.37391258804798208416416283506, −9.724834753666512757734234214155, −8.927224277887758677143991611484, −7.978160224168158140251408089036, −7.24585139091341222686434787917, −6.21542163822550895285784622637, −4.69085015213962146054810918740, −3.44043040816682199592424956479, −2.20414762625595912585295534888, −1.08157728289892526687687787713,
1.73942467731793535078407830299, 2.82208106807947248727651802339, 4.90864571542818068026150388706, 5.43150879916108653106960561769, 6.58935699956427518698356602120, 7.52203072759901661862618447846, 8.561722802061834437890212696574, 9.178894782211836771711280764618, 9.975529608294873375026793718117, 10.52932310641359733335607002899
