| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s − 3·13-s + 15-s + 4·17-s + 19-s − 21-s + 2·23-s − 4·25-s − 27-s − 29-s + 4·31-s + 33-s − 35-s − 3·37-s + 3·39-s + 4·41-s + 2·43-s − 45-s + 5·47-s + 49-s − 4·51-s + 2·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.258·15-s + 0.970·17-s + 0.229·19-s − 0.218·21-s + 0.417·23-s − 4/5·25-s − 0.192·27-s − 0.185·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s − 0.493·37-s + 0.480·39-s + 0.624·41-s + 0.304·43-s − 0.149·45-s + 0.729·47-s + 1/7·49-s − 0.560·51-s + 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.217437260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.217437260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.374970040347361656616744037914, −8.290648149742437191989772413486, −7.61976102536562313381248035170, −7.01885223763301530980997431641, −5.90014716426253928576468947816, −5.21646041052173270007187855185, −4.41926303796138030329635393421, −3.41442959652415133806519102884, −2.20691885160923582980251282707, −0.77067211129666744434397370886,
0.77067211129666744434397370886, 2.20691885160923582980251282707, 3.41442959652415133806519102884, 4.41926303796138030329635393421, 5.21646041052173270007187855185, 5.90014716426253928576468947816, 7.01885223763301530980997431641, 7.61976102536562313381248035170, 8.290648149742437191989772413486, 9.374970040347361656616744037914
