| L(s) = 1 | + (2.5 + 1.65i)3-s + (3.5 + 8.29i)9-s + 16.5i·11-s − 10·13-s − 3.31i·17-s − 7·19-s + 19.8i·23-s + (−4.99 + 26.5i)27-s − 33.1i·29-s − 42·31-s + (−27.5 + 41.4i)33-s + 40·37-s + (−25 − 16.5i)39-s + 16.5i·41-s − 50·43-s + ⋯ |
| L(s) = 1 | + (0.833 + 0.552i)3-s + (0.388 + 0.921i)9-s + 1.50i·11-s − 0.769·13-s − 0.195i·17-s − 0.368·19-s + 0.865i·23-s + (−0.185 + 0.982i)27-s − 1.14i·29-s − 1.35·31-s + (−0.833 + 1.25i)33-s + 1.08·37-s + (−0.641 − 0.425i)39-s + 0.404i·41-s − 1.16·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.681187679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681187679\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.5 - 1.65i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 16.5iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 3.31iT - 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 - 19.8iT - 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 + 42T + 961T^{2} \) |
| 37 | \( 1 - 40T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50T + 1.84e3T^{2} \) |
| 47 | \( 1 + 46.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 45T + 4.48e3T^{2} \) |
| 71 | \( 1 - 33.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35T + 5.32e3T^{2} \) |
| 79 | \( 1 + 12T + 6.24e3T^{2} \) |
| 83 | \( 1 - 69.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 70T + 9.40e3T^{2} \) |
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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
−9.627817496455228236184166857632, −9.411495144197421748274782678632, −8.198644645007495612578767850126, −7.54255892290008869582082450406, −6.83015087074590882647605748021, −5.42716667676948963775412037115, −4.61210900887605113041925916715, −3.85373928436236758895579949847, −2.62858797602719479469188752118, −1.79835144335591125553856163411,
0.41090716952418475715679719021, 1.79073258657995913979332621732, 2.92745216796747528780724412230, 3.66594795170182221513517766547, 4.91000156628214671503030135763, 6.05261800286795867662885373551, 6.77871074613956438388261575661, 7.71499656737260742558414079910, 8.427273205962865973837703451892, 9.025643895099281333153523003220
