| L(s) = 1 | − 5-s + 4·7-s − 2·13-s − 2·17-s + 4·19-s + 25-s − 2·29-s − 4·35-s − 2·37-s − 6·41-s + 8·47-s + 9·49-s + 2·53-s + 4·59-s + 10·61-s + 2·65-s − 4·67-s − 14·73-s + 16·79-s + 2·85-s − 10·89-s − 8·91-s − 4·95-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.676·35-s − 0.328·37-s − 0.937·41-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.63·73-s + 1.80·79-s + 0.216·85-s − 1.05·89-s − 0.838·91-s − 0.410·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.21804173053980, −13.63143239632803, −13.34874620182990, −12.50331395752878, −12.06525364722659, −11.69002556884623, −11.25477806930473, −10.75187700984921, −10.30424256806622, −9.598409372592782, −9.133002383643487, −8.412615885639694, −8.219949070373052, −7.532817442917649, −7.163631419606343, −6.669536540514605, −5.698227358550618, −5.314421411729053, −4.845154046548255, −4.221055534015807, −3.775558970943755, −2.909506602527096, −2.306211389101198, −1.613420235468210, −0.9839425373752766, 0,
0.9839425373752766, 1.613420235468210, 2.306211389101198, 2.909506602527096, 3.775558970943755, 4.221055534015807, 4.845154046548255, 5.314421411729053, 5.698227358550618, 6.669536540514605, 7.163631419606343, 7.532817442917649, 8.219949070373052, 8.412615885639694, 9.133002383643487, 9.598409372592782, 10.30424256806622, 10.75187700984921, 11.25477806930473, 11.69002556884623, 12.06525364722659, 12.50331395752878, 13.34874620182990, 13.63143239632803, 14.21804173053980
