| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 4·11-s − 6·13-s − 2·17-s − 8·19-s − 4·21-s + 6·23-s + 4·27-s + 2·29-s + 4·31-s − 8·33-s + 2·37-s + 12·39-s − 10·41-s − 2·43-s + 2·47-s − 3·49-s + 4·51-s + 2·53-s + 16·57-s − 2·61-s + 2·63-s − 6·67-s − 12·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 0.872·21-s + 1.25·23-s + 0.769·27-s + 0.371·29-s + 0.718·31-s − 1.39·33-s + 0.328·37-s + 1.92·39-s − 1.56·41-s − 0.304·43-s + 0.291·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s + 2.11·57-s − 0.256·61-s + 0.251·63-s − 0.733·67-s − 1.44·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.940065788699458033470615965870, −8.332838638645821456148515024672, −7.06837717057143174196449719945, −6.65562219769384853070678203854, −5.72681341166535294764307852221, −4.71417813243476363904421798760, −4.42690261596024658611815234137, −2.75972747479066660033756808110, −1.51149405336740628394851182536, 0,
1.51149405336740628394851182536, 2.75972747479066660033756808110, 4.42690261596024658611815234137, 4.71417813243476363904421798760, 5.72681341166535294764307852221, 6.65562219769384853070678203854, 7.06837717057143174196449719945, 8.332838638645821456148515024672, 8.940065788699458033470615965870
