| L(s) = 1 | − 5-s − 2·7-s − 6·13-s + 2·17-s + 4·19-s + 25-s + 2·35-s + 6·37-s − 2·43-s − 8·47-s − 3·49-s − 2·53-s − 4·59-s + 12·61-s + 6·65-s + 4·67-s + 14·73-s − 4·79-s + 14·83-s − 2·85-s − 6·89-s + 12·91-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.338·35-s + 0.986·37-s − 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.520·59-s + 1.53·61-s + 0.744·65-s + 0.488·67-s + 1.63·73-s − 0.450·79-s + 1.53·83-s − 0.216·85-s − 0.635·89-s + 1.25·91-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.85978808171236, −15.29410529735947, −14.64264562368031, −14.39796256998087, −13.62697907346320, −13.01190408911494, −12.51238430992346, −12.08335567948100, −11.51191423350998, −10.97733261051430, −10.08315531674670, −9.718497858238455, −9.426282928081616, −8.501617682139904, −7.830148163831089, −7.455937380752491, −6.768354424750665, −6.262799302293675, −5.289627713063532, −4.983198071957958, −4.126885157971591, −3.370536412531947, −2.852584462791950, −2.062510327721121, −0.9049944809847285, 0,
0.9049944809847285, 2.062510327721121, 2.852584462791950, 3.370536412531947, 4.126885157971591, 4.983198071957958, 5.289627713063532, 6.262799302293675, 6.768354424750665, 7.455937380752491, 7.830148163831089, 8.501617682139904, 9.426282928081616, 9.718497858238455, 10.08315531674670, 10.97733261051430, 11.51191423350998, 12.08335567948100, 12.51238430992346, 13.01190408911494, 13.62697907346320, 14.39796256998087, 14.64264562368031, 15.29410529735947, 15.85978808171236
