L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (2.15 + 0.587i)5-s − 0.833·7-s + (0.809 + 0.587i)8-s + (−0.107 − 2.23i)10-s + (−0.257 − 0.792i)11-s + (1.41 − 4.34i)13-s + (0.257 + 0.792i)14-s + (0.309 − 0.951i)16-s + (4.41 + 3.20i)17-s + (7.00 + 5.08i)19-s + (−2.09 + 0.792i)20-s + (−0.674 + 0.489i)22-s + (−1.09 − 3.35i)23-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.964 + 0.262i)5-s − 0.314·7-s + (0.286 + 0.207i)8-s + (−0.0340 − 0.706i)10-s + (−0.0776 − 0.238i)11-s + (0.391 − 1.20i)13-s + (0.0688 + 0.211i)14-s + (0.0772 − 0.237i)16-s + (1.06 + 0.777i)17-s + (1.60 + 1.16i)19-s + (−0.467 + 0.177i)20-s + (−0.143 + 0.104i)22-s + (−0.227 − 0.700i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32738 - 0.548198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32738 - 0.548198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.15 - 0.587i)T \) |
good | 7 | \( 1 + 0.833T + 7T^{2} \) |
| 11 | \( 1 + (0.257 + 0.792i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.41 + 4.34i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.41 - 3.20i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-7.00 - 5.08i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.09 + 3.35i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.64 + 1.92i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.26 + 6.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.576 - 1.77i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + (0.674 - 0.489i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 3.77i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.18 - 12.8i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.81 - 5.59i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.21 + 0.881i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (1.91 - 1.38i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.02 + 3.16i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.18 - 3.03i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (9.97 + 7.25i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.16 - 6.66i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (8.97 - 6.51i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.69469469394733914340946345595, −10.15232584462960545375175822246, −9.507726837457905512487557215606, −8.345017310705890849074187420962, −7.50527844083091780376952741376, −5.98181571991106552228304897673, −5.47428193446121688326208762948, −3.71246174956216500661705304440, −2.79664932788655801474686401926, −1.27187375771751555041006509621,
1.39522249770388458076991751870, 3.14320449872850306700899907657, 4.80854615056743770905059815945, 5.53245503363840911367167982868, 6.65012767468318600114230284680, 7.33084620767625068753593582233, 8.606773548918154139000142187388, 9.582847819623408303450062522764, 9.746042573534892328205627887743, 11.16332769416116875244355811864