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
−11.93759267466581413959912734778, −10.52995331553340816190651239075, −9.839965753193335947221469121783, −9.289586243207462266319325250356, −7.75563693319772417686908163587, −7.30800333539704944601414165230, −5.88134194335093433658570705058, −4.60463361327662793581007307088, −3.07354511867615853338524738632, −2.22361725506315176228157428318,
1.62225798429791865177696329075, 3.00064378951663571643162597322, 4.43628686763666556951209722787, 5.60581513154021454141333167766, 6.91979190739160976315712696301, 8.060012548689438340377908689145, 8.703058600036763972645769056276, 9.922401039694395006817206968187, 10.17857844001618904867220293149, 12.05299095690399843502875814843