| L(s) = 1 | + (−1.99 + 0.169i)2-s + (3.94 − 0.675i)4-s − 12.3i·7-s + (−7.74 + 2.01i)8-s − 11.0i·11-s − 2.82·13-s + (2.10 + 24.7i)14-s + (15.0 − 5.32i)16-s + 6.52·17-s − 27.9i·19-s + (1.87 + 22.0i)22-s − 7.90i·23-s + (5.61 − 0.477i)26-s + (−8.37 − 48.8i)28-s − 50.7·29-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0847i)2-s + (0.985 − 0.168i)4-s − 1.77i·7-s + (−0.967 + 0.251i)8-s − 1.00i·11-s − 0.216·13-s + (0.150 + 1.76i)14-s + (0.942 − 0.332i)16-s + 0.383·17-s − 1.47i·19-s + (0.0850 + 1.00i)22-s − 0.343i·23-s + (0.216 − 0.0183i)26-s + (−0.298 − 1.74i)28-s − 1.74·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.168i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6298220995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6298220995\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 - 0.169i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 12.3iT - 49T^{2} \) |
| 11 | \( 1 + 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 2.82T + 169T^{2} \) |
| 17 | \( 1 - 6.52T + 289T^{2} \) |
| 19 | \( 1 + 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 7.90iT - 529T^{2} \) |
| 29 | \( 1 + 50.7T + 841T^{2} \) |
| 31 | \( 1 - 36.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 45.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 66.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 99.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 127.T + 9.40e3T^{2} \) |
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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
−9.548832952120545799964838325120, −8.695091928564198696474057900729, −7.77167460594167669612441474240, −7.14410955855332909972980365666, −6.42014564770098519822282921239, −5.20636794779481348596882773181, −3.90028212931131660538486566027, −2.87243202135576612535627845030, −1.24641974834418542617995965760, −0.29767924311994822927728567694,
1.75981861342532403512063939682, 2.45158573783515286769080168418, 3.77366059651447332348621071774, 5.45923276571972356350986393966, 5.93144981589466747267328123931, 7.15131680981116113347372867234, 7.930195280932333996355762527708, 8.706535991019570810684978756868, 9.574335394585371686391859141826, 9.917377489343697816355320429604
