L(s) = 1 | + (1.71 − 0.262i)3-s + (1.48 − 1.67i)5-s + (−0.245 − 0.245i)7-s + (2.86 − 0.899i)9-s + (−0.879 − 1.21i)11-s + (−5.29 + 0.839i)13-s + (2.10 − 3.25i)15-s + (6.61 + 3.36i)17-s + (−3.22 + 1.04i)19-s + (−0.484 − 0.355i)21-s + (−1.61 − 0.255i)23-s + (−0.592 − 4.96i)25-s + (4.66 − 2.29i)27-s + (−0.637 + 1.96i)29-s + (2.11 + 6.51i)31-s + ⋯ |
L(s) = 1 | + (0.988 − 0.151i)3-s + (0.663 − 0.747i)5-s + (−0.0927 − 0.0927i)7-s + (0.953 − 0.299i)9-s + (−0.265 − 0.365i)11-s + (−1.46 + 0.232i)13-s + (0.542 − 0.839i)15-s + (1.60 + 0.817i)17-s + (−0.738 + 0.240i)19-s + (−0.105 − 0.0776i)21-s + (−0.335 − 0.0531i)23-s + (−0.118 − 0.992i)25-s + (0.897 − 0.441i)27-s + (−0.118 + 0.364i)29-s + (0.380 + 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79491 - 0.502309i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79491 - 0.502309i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.262i)T \) |
| 5 | \( 1 + (-1.48 + 1.67i)T \) |
good | 7 | \( 1 + (0.245 + 0.245i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.879 + 1.21i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (5.29 - 0.839i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-6.61 - 3.36i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (3.22 - 1.04i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.61 + 0.255i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.637 - 1.96i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.11 - 6.51i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.18 - 7.50i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.49 + 3.44i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (3.03 - 3.03i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.43 + 2.81i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.212 - 0.108i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (2.57 + 1.87i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.40 + 2.47i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (5.71 - 11.2i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (15.0 + 4.87i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.02 - 12.7i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (9.94 + 3.23i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.61 - 12.9i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 7.42i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.73 + 1.39i)T + (57.0 - 78.4i)T^{2} \) |
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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
−12.05299095690399843502875814843, −10.17857844001618904867220293149, −9.922401039694395006817206968187, −8.703058600036763972645769056276, −8.060012548689438340377908689145, −6.91979190739160976315712696301, −5.60581513154021454141333167766, −4.43628686763666556951209722787, −3.00064378951663571643162597322, −1.62225798429791865177696329075,
2.22361725506315176228157428318, 3.07354511867615853338524738632, 4.60463361327662793581007307088, 5.88134194335093433658570705058, 7.30800333539704944601414165230, 7.75563693319772417686908163587, 9.289586243207462266319325250356, 9.839965753193335947221469121783, 10.52995331553340816190651239075, 11.93759267466581413959912734778