| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 5·13-s − 6·17-s − 6·19-s + 6·21-s + 23-s − 5·25-s − 9·27-s − 9·29-s + 3·31-s + 8·37-s − 15·39-s + 3·41-s + 8·43-s + 7·47-s − 3·49-s + 18·51-s + 2·53-s + 18·57-s − 4·59-s + 10·61-s − 12·63-s − 8·67-s − 3·69-s + 7·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 1.38·13-s − 1.45·17-s − 1.37·19-s + 1.30·21-s + 0.208·23-s − 25-s − 1.73·27-s − 1.67·29-s + 0.538·31-s + 1.31·37-s − 2.40·39-s + 0.468·41-s + 1.21·43-s + 1.02·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s + 2.38·57-s − 0.520·59-s + 1.28·61-s − 1.51·63-s − 0.977·67-s − 0.361·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6198168344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6198168344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−9.591883153701137951726562286702, −8.886282505239482250243888040201, −7.71432987644548102492721233686, −6.64463613367634943614716603788, −6.22288961527878044675757525596, −5.65959979396183493033602052738, −4.43304023015131613807265328780, −3.85633306089834148803993781414, −2.10930080492130179024883767457, −0.59849813061905194430485021847,
0.59849813061905194430485021847, 2.10930080492130179024883767457, 3.85633306089834148803993781414, 4.43304023015131613807265328780, 5.65959979396183493033602052738, 6.22288961527878044675757525596, 6.64463613367634943614716603788, 7.71432987644548102492721233686, 8.886282505239482250243888040201, 9.591883153701137951726562286702
