L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 11-s + 2·13-s + 2·15-s − 2·19-s + 2·21-s − 25-s + 27-s + 8·29-s + 4·31-s + 33-s + 4·35-s + 6·37-s + 2·39-s − 4·41-s − 6·43-s + 2·45-s − 8·47-s − 3·49-s + 6·53-s + 2·55-s − 2·57-s + 4·59-s + 6·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.458·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.174·33-s + 0.676·35-s + 0.986·37-s + 0.320·39-s − 0.624·41-s − 0.914·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.269·55-s − 0.264·57-s + 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.013631607\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.013631607\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.908178886503231346969726044983, −8.472053423840584502037121826280, −7.71033750750391809507402037528, −6.64062999690513900479159377110, −6.06600619091595052144364526022, −5.00902753488524235700649391826, −4.26972549518203271297655093030, −3.13746693181599537651789760849, −2.13314979460988605591041122276, −1.26364654697379756961046200245,
1.26364654697379756961046200245, 2.13314979460988605591041122276, 3.13746693181599537651789760849, 4.26972549518203271297655093030, 5.00902753488524235700649391826, 6.06600619091595052144364526022, 6.64062999690513900479159377110, 7.71033750750391809507402037528, 8.472053423840584502037121826280, 8.908178886503231346969726044983