| L(s) = 1 | + (0.308 + 1.38i)2-s + (1.27 + 2.50i)3-s + (−1.80 + 0.850i)4-s + (−1.94 + 1.09i)5-s + (−3.06 + 2.53i)6-s + (−0.707 − 0.707i)7-s + (−1.73 − 2.23i)8-s + (−2.88 + 3.97i)9-s + (−2.11 − 2.34i)10-s + (−0.113 − 0.156i)11-s + (−4.44 − 3.44i)12-s + (0.311 + 1.96i)13-s + (0.758 − 1.19i)14-s + (−5.24 − 3.47i)15-s + (2.55 − 3.08i)16-s + (2.57 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (0.217 + 0.975i)2-s + (0.737 + 1.44i)3-s + (−0.904 + 0.425i)4-s + (−0.870 + 0.491i)5-s + (−1.25 + 1.03i)6-s + (−0.267 − 0.267i)7-s + (−0.612 − 0.790i)8-s + (−0.962 + 1.32i)9-s + (−0.669 − 0.742i)10-s + (−0.0342 − 0.0471i)11-s + (−1.28 − 0.995i)12-s + (0.0864 + 0.545i)13-s + (0.202 − 0.319i)14-s + (−1.35 − 0.897i)15-s + (0.638 − 0.770i)16-s + (0.625 + 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620963 - 0.988423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620963 - 0.988423i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.308 - 1.38i)T \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (-1.27 - 2.50i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (0.113 + 0.156i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.311 - 1.96i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.57 - 1.31i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.09 + 3.36i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.18 - 7.46i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (6.16 + 2.00i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.24 - 0.405i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.90 + 1.09i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.540 - 0.392i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (8.00 - 8.00i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.53 - 1.29i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (2.57 - 1.31i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.06 - 2.22i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.48 + 2.53i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.15 - 4.22i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (4.27 + 1.39i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 2.15i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.57 - 7.92i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.53 + 1.80i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 14.9i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.65 + 13.0i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.91591271687854016090484603367, −9.745588257005225073063184315833, −9.411288925757577263926749502904, −8.324253523871994082062205128325, −7.71120208668343105596551318659, −6.73659776714845225590940475504, −5.52030149023464529308054402279, −4.43400401872515504880944931495, −3.80637464833826904822115315983, −3.06772710643457147795681740335,
0.52827637770855408294461396179, 1.83241177125442848893740395694, 2.96403506432524145444059667387, 3.85263946302742520412300436792, 5.20510574406499052386530978874, 6.33474394331912572421370099998, 7.52910186790008558506599444610, 8.252066253580617657986078438377, 8.794819789031064554733874779123, 9.842683331267224612659174804480
