| L(s) = 1 | + (0.0973 + 1.41i)2-s + (0.306 + 0.204i)3-s + (−1.98 + 0.274i)4-s + (1.42 + 0.283i)5-s + (−0.258 + 0.451i)6-s + (−0.666 + 1.60i)7-s + (−0.580 − 2.76i)8-s + (−1.09 − 2.64i)9-s + (−0.261 + 2.03i)10-s + (−1.65 − 2.48i)11-s + (−0.662 − 0.321i)12-s + (5.03 − 1.00i)13-s + (−2.33 − 0.783i)14-s + (0.378 + 0.378i)15-s + (3.84 − 1.08i)16-s + (−1.55 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (0.0688 + 0.997i)2-s + (0.176 + 0.118i)3-s + (−0.990 + 0.137i)4-s + (0.637 + 0.126i)5-s + (−0.105 + 0.184i)6-s + (−0.251 + 0.608i)7-s + (−0.205 − 0.978i)8-s + (−0.365 − 0.882i)9-s + (−0.0825 + 0.644i)10-s + (−0.500 − 0.748i)11-s + (−0.191 − 0.0927i)12-s + (1.39 − 0.277i)13-s + (−0.624 − 0.209i)14-s + (0.0976 + 0.0976i)15-s + (0.962 − 0.272i)16-s + (−0.376 + 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.735082 + 0.557015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735082 + 0.557015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0973 - 1.41i)T \) |
| good | 3 | \( 1 + (-0.306 - 0.204i)T + (1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-1.42 - 0.283i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.666 - 1.60i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (1.65 + 2.48i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-5.03 + 1.00i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (1.55 - 1.55i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.0359 + 0.180i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (6.50 - 2.69i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.0389 + 0.0582i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 6.25iT - 31T^{2} \) |
| 37 | \( 1 + (1.63 - 8.20i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (6.98 - 2.89i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-9.23 + 6.17i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-7.84 + 7.84i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.21 + 1.82i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (8.31 + 1.65i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (3.29 + 2.19i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-13.1 - 8.81i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-1.13 + 2.73i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (3.74 + 9.04i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (1.58 + 1.58i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.12 + 5.64i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-14.5 - 6.02i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 5.25iT - 97T^{2} \) |
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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
−15.43222408290722069777861073949, −14.04490541061761223058086230568, −13.42489732837863493022782612663, −12.07266197800570605788127631103, −10.33873481789664600378203425406, −9.052537932740325271754151750808, −8.272871267274446368419227727137, −6.35070021688989074490099298718, −5.70125970317468555048689781816, −3.54335132756074797697884816155,
2.17001961622276395100946510300, 4.17146517872050150805026606233, 5.79692429210285800372871230825, 7.84217517049303792483641165905, 9.181161112212502357062338703202, 10.31699910922364306697232420251, 11.17074090511548598444288453629, 12.64624921725698648330132420662, 13.61367230307768598251287625683, 14.06166833991554528475177879802
