| L(s) = 1 | + 7.07i·5-s − 12·7-s + 5.65i·11-s − 8·13-s + 9.89i·17-s + 16·19-s + 39.5i·23-s − 25.0·25-s − 29.6i·29-s + 4·31-s − 84.8i·35-s + 30·37-s − 21.2i·41-s + 8·43-s + 16.9i·47-s + ⋯ |
| L(s) = 1 | + 1.41i·5-s − 1.71·7-s + 0.514i·11-s − 0.615·13-s + 0.582i·17-s + 0.842·19-s + 1.72i·23-s − 1.00·25-s − 1.02i·29-s + 0.129·31-s − 2.42i·35-s + 0.810·37-s − 0.517i·41-s + 0.186·43-s + 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.398017 + 0.768910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398017 + 0.768910i\) |
| \(L(2)\) |
|
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 | \( 1 - 7.07iT - 25T^{2} \) |
| 7 | \( 1 + 12T + 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 8T + 169T^{2} \) |
| 17 | \( 1 - 9.89iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 39.5iT - 529T^{2} \) |
| 29 | \( 1 + 29.6iT - 841T^{2} \) |
| 31 | \( 1 - 4T + 961T^{2} \) |
| 37 | \( 1 - 30T + 1.36e3T^{2} \) |
| 41 | \( 1 + 21.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 79.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14T + 3.72e3T^{2} \) |
| 67 | \( 1 - 88T + 4.48e3T^{2} \) |
| 71 | \( 1 - 28.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 80T + 5.32e3T^{2} \) |
| 79 | \( 1 + 100T + 6.24e3T^{2} \) |
| 83 | \( 1 - 130. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 148. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112T + 9.40e3T^{2} \) |
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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
−13.24735222135142667057810889499, −12.23373142095143747127762562572, −11.12482701009536299409571331328, −9.924460835725363737839603744465, −9.576884334260884225917810290604, −7.59332269596883217138421250277, −6.80481579083797546684420965608, −5.82152952893211783055835309240, −3.71884225589732074966725039280, −2.68972719173802018432960162741,
0.55045154687538789914455150589, 3.01764409599214698489808494239, 4.60555681699027270270598066899, 5.84059547680772593226015249683, 7.06766094581704811118873486666, 8.562359125977629017616364519796, 9.351726225063164252769311287994, 10.20275377197463729187005314187, 11.80143691356757460153509152577, 12.69775952928540758968425766749
