| L(s) = 1 | + 3-s − 2·4-s + 5-s + 9-s + 11-s − 2·12-s + 15-s + 4·16-s − 3·17-s + 3·19-s − 2·20-s − 23-s + 25-s + 27-s − 7·29-s − 6·31-s + 33-s − 2·36-s − 8·37-s + 2·41-s − 5·43-s − 2·44-s + 45-s + 6·47-s + 4·48-s − 3·51-s − 3·53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.258·15-s + 16-s − 0.727·17-s + 0.688·19-s − 0.447·20-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.29·29-s − 1.07·31-s + 0.174·33-s − 1/3·36-s − 1.31·37-s + 0.312·41-s − 0.762·43-s − 0.301·44-s + 0.149·45-s + 0.875·47-s + 0.577·48-s − 0.420·51-s − 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.45191448637849328846141848778, −7.00187747037114153455786347387, −5.91188259672424987039908508508, −5.39415557590842578912884071561, −4.59975965109527404132462934403, −3.84552026512569092275224062018, −3.27379593060435083218630050808, −2.16908917064708392910543687415, −1.33973019606870352894302817616, 0,
1.33973019606870352894302817616, 2.16908917064708392910543687415, 3.27379593060435083218630050808, 3.84552026512569092275224062018, 4.59975965109527404132462934403, 5.39415557590842578912884071561, 5.91188259672424987039908508508, 7.00187747037114153455786347387, 7.45191448637849328846141848778
