L(s) = 1 | + (0.951 + 0.309i)3-s + (−1.98 + 1.02i)5-s + 3.54i·7-s + (0.809 + 0.587i)9-s + (1.78 − 1.29i)11-s + (−4.21 + 5.80i)13-s + (−2.20 + 0.358i)15-s + (6.05 − 1.96i)17-s + (0.715 + 2.20i)19-s + (−1.09 + 3.37i)21-s + (−1.27 − 1.76i)23-s + (2.90 − 4.06i)25-s + (0.587 + 0.809i)27-s + (−0.262 + 0.806i)29-s + (−1.32 − 4.09i)31-s + ⋯ |
L(s) = 1 | + (0.549 + 0.178i)3-s + (−0.889 + 0.457i)5-s + 1.34i·7-s + (0.269 + 0.195i)9-s + (0.538 − 0.390i)11-s + (−1.17 + 1.61i)13-s + (−0.569 + 0.0925i)15-s + (1.46 − 0.477i)17-s + (0.164 + 0.504i)19-s + (−0.239 + 0.736i)21-s + (−0.266 − 0.367i)23-s + (0.581 − 0.813i)25-s + (0.113 + 0.155i)27-s + (−0.0486 + 0.149i)29-s + (−0.238 − 0.734i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.984983 + 0.821503i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.984983 + 0.821503i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (1.98 - 1.02i)T \) |
good | 7 | \( 1 - 3.54iT - 7T^{2} \) |
| 11 | \( 1 + (-1.78 + 1.29i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (4.21 - 5.80i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-6.05 + 1.96i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.715 - 2.20i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.27 + 1.76i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.262 - 0.806i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.32 + 4.09i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.24 + 5.84i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 0.790i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.18iT - 43T^{2} \) |
| 47 | \( 1 + (5.75 + 1.87i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-11.3 - 3.69i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (10.0 + 7.33i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.59 + 4.06i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.50 + 1.46i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.25 + 13.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.640 + 0.881i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.80 + 5.55i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (11.9 - 3.87i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (5.68 - 4.12i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-17.2 - 5.60i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.94501757037074332162832078784, −11.28538893746646122523295418274, −9.792275616378617615762385604476, −9.192958796325453151250011197669, −8.137079052661540020638030259476, −7.27857962042715345875066838348, −6.07212077052436709889662788141, −4.69844002929040538105253177368, −3.48789650025403627787696165306, −2.29631494461843530007588277442,
0.961941630125111895059538128162, 3.18037495476153487062494803698, 4.13428885632573439943912224912, 5.32006591786436844908980156520, 7.14614905013750159404745765907, 7.60217866388042628815678030148, 8.442685726087277766317344128250, 9.831324334320860003782044471049, 10.38140774577772311990165233250, 11.70296713961879175729411434229