| L(s) = 1 | + 3-s − 2·9-s − 11-s + 2·13-s + 3·17-s + 3·19-s + 6·23-s − 5·27-s + 2·29-s + 2·31-s − 33-s − 4·37-s + 2·39-s − 3·41-s − 4·43-s − 6·47-s − 7·49-s + 3·51-s + 10·53-s + 3·57-s + 12·59-s + 12·61-s − 67-s + 6·69-s + 10·71-s + 73-s + 16·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.688·19-s + 1.25·23-s − 0.962·27-s + 0.371·29-s + 0.359·31-s − 0.174·33-s − 0.657·37-s + 0.320·39-s − 0.468·41-s − 0.609·43-s − 0.875·47-s − 49-s + 0.420·51-s + 1.37·53-s + 0.397·57-s + 1.56·59-s + 1.53·61-s − 0.122·67-s + 0.722·69-s + 1.18·71-s + 0.117·73-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.239829147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.239829147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−8.461322813924786344237643647339, −8.195058002925843492219111025841, −7.21589563736530399110295158414, −6.50218780484881389219653643120, −5.47548489801813724488423091686, −4.99955466497131575988512939589, −3.64269076679621170829736825808, −3.17062287784593562357719197405, −2.18128920290272584123860382730, −0.889257867698733818053379217702,
0.889257867698733818053379217702, 2.18128920290272584123860382730, 3.17062287784593562357719197405, 3.64269076679621170829736825808, 4.99955466497131575988512939589, 5.47548489801813724488423091686, 6.50218780484881389219653643120, 7.21589563736530399110295158414, 8.195058002925843492219111025841, 8.461322813924786344237643647339
