| L(s) = 1 | + 13.8i·7-s + 22·13-s + 27.7i·19-s − 25·25-s − 41.5i·31-s + 26·37-s − 83.1i·43-s − 142.·49-s + 74·61-s − 55.4i·67-s + 46·73-s + 69.2i·79-s + 304. i·91-s − 2·97-s − 69.2i·103-s + ⋯ |
| L(s) = 1 | + 1.97i·7-s + 1.69·13-s + 1.45i·19-s − 25-s − 1.34i·31-s + 0.702·37-s − 1.93i·43-s − 2.91·49-s + 1.21·61-s − 0.827i·67-s + 0.630·73-s + 0.876i·79-s + 3.34i·91-s − 0.0206·97-s − 0.672i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22747 + 0.708682i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22747 + 0.708682i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 - 13.8iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 22T + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 27.7iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 + 41.5iT - 961T^{2} \) |
| 37 | \( 1 - 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 83.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.2iT - 6.24e3T^{2} \) |
| 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
−12.90539898338966273385585731390, −11.99136602115939162663967842668, −11.23120992621696129989448378933, −9.821688481150143316916365891363, −8.786163962267527742721346625241, −8.048676287381564243442379648000, −6.15467344300203995936378882504, −5.60334582928723523766446307989, −3.70805132609599974370180741838, −2.05672870820317388577703022384,
1.03888745904031585352193471699, 3.49873381301590419280955937827, 4.56917667536872403071352635563, 6.33517829855050870100216406095, 7.29681091880988415773949374646, 8.411678163774771152296237552502, 9.741938197893628161604984894299, 10.80797365925877133173254232423, 11.34422238443918180363199345699, 13.09403540652120358063424147791
