| L(s) = 1 | + 3·5-s + 7-s − 2·13-s + 3·17-s + 4·25-s + 6·29-s − 6·31-s + 3·35-s − 37-s − 9·41-s − 7·43-s + 7·47-s + 49-s + 2·53-s − 5·59-s − 8·61-s − 6·65-s + 7·67-s − 8·71-s − 6·73-s + 3·79-s − 7·83-s + 9·85-s + 17·89-s − 2·91-s − 18·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.554·13-s + 0.727·17-s + 4/5·25-s + 1.11·29-s − 1.07·31-s + 0.507·35-s − 0.164·37-s − 1.40·41-s − 1.06·43-s + 1.02·47-s + 1/7·49-s + 0.274·53-s − 0.650·59-s − 1.02·61-s − 0.744·65-s + 0.855·67-s − 0.949·71-s − 0.702·73-s + 0.337·79-s − 0.768·83-s + 0.976·85-s + 1.80·89-s − 0.209·91-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.19139825405232, −12.42397262839549, −12.18970643412961, −11.78964500701590, −11.03812062339002, −10.68251763814326, −10.10964245664103, −9.903792291904003, −9.427112220346960, −8.796456366322738, −8.544475820007356, −7.838488434073241, −7.387361427062212, −6.847864456077323, −6.399550757294111, −5.763654723303982, −5.502783802706327, −4.911612838918568, −4.557675586072824, −3.737583355818076, −3.148370304199814, −2.647719694080028, −1.938227324581327, −1.628145952150994, −0.9328746800354305, 0,
0.9328746800354305, 1.628145952150994, 1.938227324581327, 2.647719694080028, 3.148370304199814, 3.737583355818076, 4.557675586072824, 4.911612838918568, 5.502783802706327, 5.763654723303982, 6.399550757294111, 6.847864456077323, 7.387361427062212, 7.838488434073241, 8.544475820007356, 8.796456366322738, 9.427112220346960, 9.903792291904003, 10.10964245664103, 10.68251763814326, 11.03812062339002, 11.78964500701590, 12.18970643412961, 12.42397262839549, 13.19139825405232
