| L(s) = 1 | + 2.82i·3-s − 5-s − 5·7-s + 0.999·9-s − 5·11-s + 16.9i·13-s − 2.82i·15-s − 25·17-s − 19·19-s − 14.1i·21-s + 10·23-s − 24·25-s + 28.2i·27-s − 42.4i·29-s − 42.4i·31-s + ⋯ |
| L(s) = 1 | + 0.942i·3-s − 0.200·5-s − 0.714·7-s + 0.111·9-s − 0.454·11-s + 1.30i·13-s − 0.188i·15-s − 1.47·17-s − 19-s − 0.673i·21-s + 0.434·23-s − 0.959·25-s + 1.04i·27-s − 1.46i·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.603443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.603443i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 2.82iT - 9T^{2} \) |
| 5 | \( 1 + T + 25T^{2} \) |
| 7 | \( 1 + 5T + 49T^{2} \) |
| 11 | \( 1 + 5T + 121T^{2} \) |
| 13 | \( 1 - 16.9iT - 169T^{2} \) |
| 17 | \( 1 + 25T + 289T^{2} \) |
| 23 | \( 1 - 10T + 529T^{2} \) |
| 29 | \( 1 + 42.4iT - 841T^{2} \) |
| 31 | \( 1 + 42.4iT - 961T^{2} \) |
| 37 | \( 1 - 25.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 95T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 130T + 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.9iT - 9.40e3T^{2} \) |
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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
−11.70832563362660317317337754457, −11.02125266818162783581775244859, −9.929832168873740650870943323342, −9.399008423768557816554457060099, −8.358246437050222303638081471313, −7.00215994013414804572339390639, −6.10112240731783460707151321019, −4.52428916598865665784564564396, −4.00350686533231408432810097176, −2.34385693728191459883639534368,
0.27301913763189484615721775473, 2.09586188986499122709036501009, 3.48007248275572484484663990982, 5.02011616635364706586195241331, 6.34664494505104264980815548554, 7.05652417254592547482618923282, 8.050747824597529460200638406356, 8.977335498184274514108441299337, 10.27195752093586346980793830614, 10.95794059830453745703831634225
