L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s + 2·19-s − 4·21-s − 6·23-s + 25-s + 27-s − 6·29-s − 10·31-s + 4·33-s + 4·35-s − 37-s + 2·39-s − 2·41-s + 2·43-s − 45-s + 4·47-s + 9·49-s + 2·51-s + 10·53-s − 4·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.458·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.696·33-s + 0.676·35-s − 0.164·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.21926630378350476268335375684, −6.96508549557073085237015466049, −5.98084639911729352920572736576, −5.61026164758803449801977151336, −4.20611857983035973809175420302, −3.69238914280008987422092049833, −3.36269887825618013694118670117, −2.27366356695913427981582014377, −1.25162873873077222773873005415, 0,
1.25162873873077222773873005415, 2.27366356695913427981582014377, 3.36269887825618013694118670117, 3.69238914280008987422092049833, 4.20611857983035973809175420302, 5.61026164758803449801977151336, 5.98084639911729352920572736576, 6.96508549557073085237015466049, 7.21926630378350476268335375684