| L(s) = 1 | + (0.438 − 1.95i)2-s + (−3.61 − 1.71i)4-s + 6.33i·7-s + (−4.92 + 6.30i)8-s − 9.27i·11-s − 18.5·13-s + (12.3 + 2.77i)14-s + (10.1 + 12.3i)16-s + 13.9·17-s − 17.2i·19-s + (−18.1 − 4.06i)22-s + 33.7i·23-s + (−8.13 + 36.2i)26-s + (10.8 − 22.8i)28-s + 28.6·29-s + ⋯ |
| L(s) = 1 | + (0.219 − 0.975i)2-s + (−0.904 − 0.427i)4-s + 0.904i·7-s + (−0.615 + 0.788i)8-s − 0.843i·11-s − 1.42·13-s + (0.882 + 0.198i)14-s + (0.634 + 0.772i)16-s + 0.818·17-s − 0.907i·19-s + (−0.823 − 0.184i)22-s + 1.46i·23-s + (−0.312 + 1.39i)26-s + (0.386 − 0.817i)28-s + 0.986·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.587985524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587985524\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.438 + 1.95i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6.33iT - 49T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 - 13.9T + 289T^{2} \) |
| 19 | \( 1 + 17.2iT - 361T^{2} \) |
| 23 | \( 1 - 33.7iT - 529T^{2} \) |
| 29 | \( 1 - 28.6T + 841T^{2} \) |
| 31 | \( 1 - 23.4iT - 961T^{2} \) |
| 37 | \( 1 - 67.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.49iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 13.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 69.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 68.5T + 9.40e3T^{2} \) |
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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
−9.743877581229600254418776868881, −9.336827001782902360246783968088, −8.394496376736259454016087668354, −7.47190184547941381264734512658, −6.04573951221701606588590214725, −5.33974333522932429602403814697, −4.46882594988423408612062243381, −3.10075531439049448464595550376, −2.50120411869075179841576509696, −0.979056111364848870891410271759,
0.62543665014783535672276501322, 2.58155328479243912707798848624, 4.05261009313190974681773778322, 4.60441386103850706282214439463, 5.66108298524738991664029054962, 6.66726106521028386262720343047, 7.47165071617801483590739787141, 7.900077311743655583967655348670, 9.063677039254514734615778637740, 10.04362870674439272596334857785
