L(s) = 1 | + 2·13-s − 19-s − 5·25-s − 7·31-s − 10·37-s + 5·43-s − 61-s − 16·67-s + 17·73-s − 4·79-s − 19·97-s + 20·103-s + 17·109-s + ⋯ |
L(s) = 1 | + 0.554·13-s − 0.229·19-s − 25-s − 1.25·31-s − 1.64·37-s + 0.762·43-s − 0.128·61-s − 1.95·67-s + 1.98·73-s − 0.450·79-s − 1.92·97-s + 1.97·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.77292717285844287077430606224, −7.19987656142895291490537858783, −6.34611962580097835376042656994, −5.70672942858072608683987301566, −4.96186526924579305811934690200, −3.99020255824923125365243219906, −3.41994640901582669879597213521, −2.29915646582559944641729960101, −1.42027017924672195504199469278, 0,
1.42027017924672195504199469278, 2.29915646582559944641729960101, 3.41994640901582669879597213521, 3.99020255824923125365243219906, 4.96186526924579305811934690200, 5.70672942858072608683987301566, 6.34611962580097835376042656994, 7.19987656142895291490537858783, 7.77292717285844287077430606224