| L(s) = 1 | + i·3-s + 2.82i·5-s − 9-s − i·11-s + 0.635i·13-s − 2.82·15-s + 2.89·17-s + 4.89i·19-s − 0.635·23-s − 3.00·25-s − i·27-s + 6.29i·29-s − 2.19·31-s + 33-s + 6.92i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.26i·5-s − 0.333·9-s − 0.301i·11-s + 0.176i·13-s − 0.730·15-s + 0.703·17-s + 1.12i·19-s − 0.132·23-s − 0.600·25-s − 0.192i·27-s + 1.16i·29-s − 0.393·31-s + 0.174·33-s + 1.13i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243447141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243447141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 0.635iT - 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 0.635T + 23T^{2} \) |
| 29 | \( 1 - 6.29iT - 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 0.898iT - 43T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 - 4.09iT - 53T^{2} \) |
| 59 | \( 1 + 13.7iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 5.79iT - 67T^{2} \) |
| 71 | \( 1 - 5.02T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 9.79iT - 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.698728494298391130131527242438, −8.655961990803140411863984364890, −7.916196315118832022138024030714, −7.07244742457713121376210028974, −6.30728026013210624755919889112, −5.57228720593655138499931268198, −4.56696141110462261246993809392, −3.41808033391836452614725319320, −3.08138725606455416524843589333, −1.66758245912943669175196845977,
0.43920816540988935868016217941, 1.51374162101673647367645369940, 2.62347172807008920492244998320, 3.88201011576000596278030310689, 4.84002200336604252111744900564, 5.46646938847989204631444798507, 6.37414515233415171640358463560, 7.31883337080180286273818899782, 7.990955646083513676081263741834, 8.710525697101786171352038423965
