L(s) = 1 | + (2.91 − 0.708i)3-s + (−4.96 + 8.59i)7-s + (7.99 − 4.12i)9-s + (3.59 + 2.07i)11-s + (−0.613 − 1.06i)13-s + 21.0i·17-s − 9.84·19-s + (−8.38 + 28.5i)21-s + (2.32 − 1.34i)23-s + (20.3 − 17.6i)27-s + (−0.321 − 0.185i)29-s + (22.1 + 38.4i)31-s + (11.9 + 3.50i)33-s − 33.0·37-s + (−2.53 − 2.66i)39-s + ⋯ |
L(s) = 1 | + (0.971 − 0.236i)3-s + (−0.708 + 1.22i)7-s + (0.888 − 0.458i)9-s + (0.326 + 0.188i)11-s + (−0.0471 − 0.0817i)13-s + 1.23i·17-s − 0.517·19-s + (−0.399 + 1.36i)21-s + (0.100 − 0.0582i)23-s + (0.755 − 0.655i)27-s + (−0.0110 − 0.00640i)29-s + (0.715 + 1.23i)31-s + (0.361 + 0.106i)33-s − 0.893·37-s + (−0.0651 − 0.0682i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.193931952\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.193931952\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.91 + 0.708i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (4.96 - 8.59i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.59 - 2.07i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (0.613 + 1.06i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 21.0iT - 289T^{2} \) |
| 19 | \( 1 + 9.84T + 361T^{2} \) |
| 23 | \( 1 + (-2.32 + 1.34i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (0.321 + 0.185i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-22.1 - 38.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 33.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (34.2 - 19.7i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (14.5 - 25.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-68.6 - 39.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 92.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (53.3 - 30.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-12.6 + 21.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-17.9 - 31.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 62.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-71.9 + 124. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (14.2 + 8.20i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 155. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (52.3 - 90.6i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.948758205942813086803017542049, −8.976139428235741157908574128668, −8.677767486354693376630458766777, −7.69235598353057144113433688931, −6.60290209243787658100563977365, −5.99957414725083822959617513677, −4.64441589000443507488521183376, −3.49871863410090690759635520335, −2.65042841336486185508883232036, −1.56466500627473729904824421260,
0.62646754899912878357522714102, 2.22778549934075934785021598048, 3.42652244715687296518268802658, 4.05367375632900908767603345882, 5.12638041318909674563462777712, 6.65135504461592436899533542821, 7.14832027640051293736749677729, 8.076809740645782658336184260258, 8.974516772037982661897743281442, 9.766345752428768138937674198252