| L(s) = 1 | + 3i·3-s − 2·7-s − 9·9-s + 22i·13-s − 26i·19-s − 6i·21-s − 25·25-s − 27i·27-s − 46·31-s + 26i·37-s − 66·39-s − 22i·43-s − 45·49-s + 78·57-s − 74i·61-s + ⋯ |
| L(s) = 1 | + i·3-s − 0.285·7-s − 9-s + 1.69i·13-s − 1.36i·19-s − 0.285i·21-s − 25-s − i·27-s − 1.48·31-s + 0.702i·37-s − 1.69·39-s − 0.511i·43-s − 0.918·49-s + 1.36·57-s − 1.21i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2247697236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2247697236\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 22iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 26iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 + 46T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 22iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 74iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 122iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 142T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−10.69901461569784638289812565629, −9.458238681259721737107403775751, −9.357660586209115886535279899535, −8.310134953272945720365024022204, −7.08546330163177382204926681885, −6.25025281573276720863993967569, −5.12016377703700012228855489155, −4.30381774664298985482186582371, −3.38044693374623922639561692672, −2.06779576055598114317599754321,
0.07421213470976312642838662138, 1.49925414073182010686894721462, 2.79309353029475354324169064819, 3.79827736007688958404857099721, 5.53542024653655314612639049616, 5.88484795060959179149772808357, 7.11529539740192100346029068721, 7.83813511511653399244683922728, 8.465810867008295580775212074127, 9.622497719935074179156296211670
