L(s) = 1 | + 3-s − 7-s + 9-s − 2·11-s − 2·13-s − 2·19-s − 21-s + 8·23-s + 27-s + 2·29-s + 6·31-s − 2·33-s − 8·37-s − 2·39-s − 10·41-s − 12·47-s + 49-s + 2·53-s − 2·57-s + 2·61-s − 63-s + 4·67-s + 8·69-s − 14·71-s + 2·73-s + 2·77-s − 4·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.458·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.348·33-s − 1.31·37-s − 0.320·39-s − 1.56·41-s − 1.75·47-s + 1/7·49-s + 0.274·53-s − 0.264·57-s + 0.256·61-s − 0.125·63-s + 0.488·67-s + 0.963·69-s − 1.66·71-s + 0.234·73-s + 0.227·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−7.41039143132492563465345332977, −6.84577161009535595755240813706, −6.25134953548335880054479102348, −5.04743006599506087557612684225, −4.87295231769059715797430599127, −3.69587330270183549997447075599, −3.04098011941302500540381330875, −2.38627311377974853038518585288, −1.33329755265638165419938925923, 0,
1.33329755265638165419938925923, 2.38627311377974853038518585288, 3.04098011941302500540381330875, 3.69587330270183549997447075599, 4.87295231769059715797430599127, 5.04743006599506087557612684225, 6.25134953548335880054479102348, 6.84577161009535595755240813706, 7.41039143132492563465345332977