| L(s) = 1 | + 3-s + 3·5-s − 2·7-s − 2·9-s − 11-s + 4·13-s + 3·15-s + 6·17-s + 8·19-s − 2·21-s + 3·23-s + 4·25-s − 5·27-s − 5·31-s − 33-s − 6·35-s + 37-s + 4·39-s − 10·43-s − 6·45-s − 3·49-s + 6·51-s + 6·53-s − 3·55-s + 8·57-s + 3·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.774·15-s + 1.45·17-s + 1.83·19-s − 0.436·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.898·31-s − 0.174·33-s − 1.01·35-s + 0.164·37-s + 0.640·39-s − 1.52·43-s − 0.894·45-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.404·55-s + 1.05·57-s + 0.390·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159553409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159553409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.11903216163362853339678805280, −9.610210113213273792621314474885, −8.901169398745525002958448579942, −7.948144712624414981153354525702, −6.87940646463052539617018989539, −5.72427671804102205390312745209, −5.44949075059971672760817509463, −3.47350326380949607246521494930, −2.88597193878705003332421314931, −1.40865503128588890763537566425,
1.40865503128588890763537566425, 2.88597193878705003332421314931, 3.47350326380949607246521494930, 5.44949075059971672760817509463, 5.72427671804102205390312745209, 6.87940646463052539617018989539, 7.948144712624414981153354525702, 8.901169398745525002958448579942, 9.610210113213273792621314474885, 10.11903216163362853339678805280
