| L(s) = 1 | + 2·5-s − 4·11-s + 6·13-s + 2·17-s − 4·19-s − 8·23-s − 25-s − 2·29-s + 10·37-s − 6·41-s + 4·43-s + 6·53-s − 8·55-s − 4·59-s + 6·61-s + 12·65-s − 4·67-s − 8·71-s − 10·73-s + 4·83-s + 4·85-s − 6·89-s − 8·95-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.64·37-s − 0.937·41-s + 0.609·43-s + 0.824·53-s − 1.07·55-s − 0.520·59-s + 0.768·61-s + 1.48·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.439·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.59825801356115, −14.90976032313710, −14.38009940692559, −13.72923332620193, −13.32679034686994, −13.07653269687194, −12.36072362273642, −11.69623912790644, −11.08752063337224, −10.57158085352897, −10.05842511579625, −9.713495758100589, −8.802877182003785, −8.443200974906607, −7.830586419754088, −7.285145742757054, −6.196397506640142, −6.044460599937452, −5.606001819538911, −4.708803057303018, −4.028808521337184, −3.392288874489316, −2.479917686562370, −1.973120405548112, −1.144793199098371, 0,
1.144793199098371, 1.973120405548112, 2.479917686562370, 3.392288874489316, 4.028808521337184, 4.708803057303018, 5.606001819538911, 6.044460599937452, 6.196397506640142, 7.285145742757054, 7.830586419754088, 8.443200974906607, 8.802877182003785, 9.713495758100589, 10.05842511579625, 10.57158085352897, 11.08752063337224, 11.69623912790644, 12.36072362273642, 13.07653269687194, 13.32679034686994, 13.72923332620193, 14.38009940692559, 14.90976032313710, 15.59825801356115
