| L(s) = 1 | + (−0.635 − 1.89i)2-s + (−3.19 + 2.40i)4-s + 10.1·7-s + (6.59 + 4.52i)8-s + 10.6i·11-s + 11.7i·13-s + (−6.41 − 19.1i)14-s + (4.38 − 15.3i)16-s + 16.6i·17-s − 0.464i·19-s + (20.2 − 6.77i)22-s − 42.1·23-s + (22.3 − 7.47i)26-s + (−32.2 + 24.3i)28-s − 19.5·29-s + ⋯ |
| L(s) = 1 | + (−0.317 − 0.948i)2-s + (−0.798 + 0.602i)4-s + 1.44·7-s + (0.824 + 0.565i)8-s + 0.968i·11-s + 0.905i·13-s + (−0.458 − 1.36i)14-s + (0.274 − 0.961i)16-s + 0.978i·17-s − 0.0244i·19-s + (0.918 − 0.307i)22-s − 1.83·23-s + (0.858 − 0.287i)26-s + (−1.15 + 0.869i)28-s − 0.674·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.085258982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085258982\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.635 + 1.89i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10.1T + 49T^{2} \) |
| 11 | \( 1 - 10.6iT - 121T^{2} \) |
| 13 | \( 1 - 11.7iT - 169T^{2} \) |
| 17 | \( 1 - 16.6iT - 289T^{2} \) |
| 19 | \( 1 + 0.464iT - 361T^{2} \) |
| 23 | \( 1 + 42.1T + 529T^{2} \) |
| 29 | \( 1 + 19.5T + 841T^{2} \) |
| 31 | \( 1 - 9.17iT - 961T^{2} \) |
| 37 | \( 1 + 23.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 58.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 41.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 63.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 22.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 61.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 15iT - 9.40e3T^{2} \) |
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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
−10.12951447927363315210410717355, −9.384657426040469579281246759967, −8.331406499420630949394500889557, −7.937904647692694727903949648126, −6.82581861708134146907992430486, −5.37834684130638250412066195428, −4.47555558198222868876323337694, −3.80658164463570579421698843596, −2.05997246259966880450454194778, −1.63657417010916430621227509663,
0.39240619602723147083933008887, 1.77882693223152179086709996203, 3.53511776154371110287719360708, 4.78243854464040521964320212653, 5.40689617408610981004465036449, 6.28178693105381654828182893592, 7.41991005836479671526976907256, 8.176561286424336411861816195307, 8.491752122480242658561963199225, 9.699567289384589612781410794682
