| L(s) = 1 | − 14.8·3-s + 166.·7-s − 21.7·9-s + 121·11-s + 491.·13-s − 1.58e3·17-s + 172.·19-s − 2.47e3·21-s − 2.34e3·23-s + 3.93e3·27-s + 3.71e3·29-s + 7.94e3·31-s − 1.79e3·33-s + 3.46e3·37-s − 7.31e3·39-s + 1.25e4·41-s + 1.92e4·43-s − 2.76e4·47-s + 1.09e4·49-s + 2.35e4·51-s + 3.32e4·53-s − 2.56e3·57-s − 3.80e4·59-s − 6.56e3·61-s − 3.62e3·63-s − 6.00e4·67-s + 3.49e4·69-s + ⋯ |
| L(s) = 1 | − 0.954·3-s + 1.28·7-s − 0.0895·9-s + 0.301·11-s + 0.807·13-s − 1.32·17-s + 0.109·19-s − 1.22·21-s − 0.926·23-s + 1.03·27-s + 0.819·29-s + 1.48·31-s − 0.287·33-s + 0.416·37-s − 0.770·39-s + 1.16·41-s + 1.58·43-s − 1.82·47-s + 0.649·49-s + 1.26·51-s + 1.62·53-s − 0.104·57-s − 1.42·59-s − 0.225·61-s − 0.115·63-s − 1.63·67-s + 0.883·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.750521012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750521012\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 + 14.8T + 243T^{2} \) |
| 7 | \( 1 - 166.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 491.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 172.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.76e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.56e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.00e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.45e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.98e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.29e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.64e5T + 8.58e9T^{2} \) |
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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
−8.929704669057669938034998743683, −8.385321088091424797728130935036, −7.47864692820783619115644707375, −6.30461113518017143938901629316, −5.91978356561965489492213650809, −4.70412306344966319129672104691, −4.31211441432061659396914629450, −2.73854356549439115407926822824, −1.56549028518069819016539282236, −0.62673361611550052738025976653,
0.62673361611550052738025976653, 1.56549028518069819016539282236, 2.73854356549439115407926822824, 4.31211441432061659396914629450, 4.70412306344966319129672104691, 5.91978356561965489492213650809, 6.30461113518017143938901629316, 7.47864692820783619115644707375, 8.385321088091424797728130935036, 8.929704669057669938034998743683
