L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s + 4·11-s − 2·15-s − 6·17-s + 2·19-s − 2·21-s − 4·23-s − 25-s − 27-s + 6·29-s + 2·31-s − 4·33-s + 4·35-s + 8·37-s − 6·41-s − 4·43-s + 2·45-s + 8·47-s − 3·49-s + 6·51-s + 2·53-s + 8·55-s − 2·57-s + 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.696·33-s + 0.676·35-s + 1.31·37-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.840·51-s + 0.274·53-s + 1.07·55-s − 0.264·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.408609198\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.408609198\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.83951771480179276370107503714, −6.88145231466281228812332353630, −6.39261061797726120649050736622, −5.88377432257670444126010988713, −4.97355221671789948126284106482, −4.46479261574920307220031232038, −3.66922686045540868903156467236, −2.37747740119356139666801790757, −1.76164770350939731567156021553, −0.822430023137796435172149883586,
0.822430023137796435172149883586, 1.76164770350939731567156021553, 2.37747740119356139666801790757, 3.66922686045540868903156467236, 4.46479261574920307220031232038, 4.97355221671789948126284106482, 5.88377432257670444126010988713, 6.39261061797726120649050736622, 6.88145231466281228812332353630, 7.83951771480179276370107503714