| L(s) = 1 | + (−1.84 − 2.36i)3-s − 2.88i·5-s − 6.13·7-s + (−2.16 + 8.73i)9-s − 14.8·13-s + (−6.81 + 5.32i)15-s + 19.3i·17-s + 6.85·19-s + (11.3 + 14.5i)21-s + 30.2i·23-s + 16.6·25-s + (24.6 − 11.0i)27-s − 21.9i·29-s + 23.8·31-s + 17.6i·35-s + ⋯ |
| L(s) = 1 | + (−0.616 − 0.787i)3-s − 0.576i·5-s − 0.876·7-s + (−0.241 + 0.970i)9-s − 1.14·13-s + (−0.454 + 0.355i)15-s + 1.13i·17-s + 0.360·19-s + (0.540 + 0.690i)21-s + 1.31i·23-s + 0.667·25-s + (0.912 − 0.407i)27-s − 0.757i·29-s + 0.769·31-s + 0.505i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.017270860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017270860\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.84 + 2.36i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2.88iT - 25T^{2} \) |
| 7 | \( 1 + 6.13T + 49T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 - 19.3iT - 289T^{2} \) |
| 19 | \( 1 - 6.85T + 361T^{2} \) |
| 23 | \( 1 - 30.2iT - 529T^{2} \) |
| 29 | \( 1 + 21.9iT - 841T^{2} \) |
| 31 | \( 1 - 23.8T + 961T^{2} \) |
| 37 | \( 1 - 28.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 72.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 22.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.61iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 115. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 75.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 41.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 83.3T + 9.40e3T^{2} \) |
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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
−9.260954355465939688517893668616, −8.289048802921821797134463574862, −7.50782963569456435489794054940, −6.78835449244143597104442548083, −5.91656926390629208947485712449, −5.25216017728955777456538629888, −4.22418675337641888561736889243, −2.94226128784853894510944590241, −1.78890261159792338230435080128, −0.56027994381556640420737081501,
0.60410413151163158321769679931, 2.70985752747790845613546328527, 3.24880736253625006076675634580, 4.60032546372697816961216690821, 5.08595040584155274564786029235, 6.41018797565752351426845065179, 6.67225535392616897200999036271, 7.73699517925549852200458319302, 8.924042529315305960243306846683, 9.760745311208194742401617153457
