| L(s) = 1 | + 5-s − 13-s + 6·17-s − 4·23-s + 25-s + 10·29-s − 6·37-s − 2·41-s + 4·43-s − 7·49-s + 6·53-s + 6·61-s − 65-s − 4·67-s + 16·71-s − 2·73-s + 4·83-s + 6·85-s + 6·89-s + 14·97-s − 6·101-s − 12·103-s − 4·107-s − 14·109-s − 10·113-s − 4·115-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.277·13-s + 1.45·17-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.768·61-s − 0.124·65-s − 0.488·67-s + 1.89·71-s − 0.234·73-s + 0.439·83-s + 0.650·85-s + 0.635·89-s + 1.42·97-s − 0.597·101-s − 1.18·103-s − 0.386·107-s − 1.34·109-s − 0.940·113-s − 0.373·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351259045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351259045\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.86024313710614696532097430664, −6.90823153056793002056497471386, −6.41051413056067833990175014344, −5.55860505151888914144681124469, −5.10592887722971602042874207321, −4.20725491653826331663773790383, −3.38062750090213723463354695459, −2.63741599748730669205673807869, −1.71741878316152659310548007340, −0.75375237820186639289019479057,
0.75375237820186639289019479057, 1.71741878316152659310548007340, 2.63741599748730669205673807869, 3.38062750090213723463354695459, 4.20725491653826331663773790383, 5.10592887722971602042874207321, 5.55860505151888914144681124469, 6.41051413056067833990175014344, 6.90823153056793002056497471386, 7.86024313710614696532097430664
