L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 2·13-s − 8·17-s − 2·19-s − 21-s − 23-s − 27-s + 29-s + 6·31-s + 33-s − 9·37-s − 2·39-s + 43-s − 6·47-s + 49-s + 8·51-s − 2·53-s + 2·57-s − 6·59-s + 8·61-s + 63-s + 3·67-s + 69-s + 7·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.94·17-s − 0.458·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.185·29-s + 1.07·31-s + 0.174·33-s − 1.47·37-s − 0.320·39-s + 0.152·43-s − 0.875·47-s + 1/7·49-s + 1.12·51-s − 0.274·53-s + 0.264·57-s − 0.781·59-s + 1.02·61-s + 0.125·63-s + 0.366·67-s + 0.120·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.609504417396998023119856255437, −8.092912834085306122697436594710, −6.89419420223668771721570585118, −6.48844234539210456547561997154, −5.49862489363903650710448281537, −4.67090121857049466764518806691, −3.97926271718894819076907901901, −2.63600659905381128296820684234, −1.56247941740261692810376944669, 0,
1.56247941740261692810376944669, 2.63600659905381128296820684234, 3.97926271718894819076907901901, 4.67090121857049466764518806691, 5.49862489363903650710448281537, 6.48844234539210456547561997154, 6.89419420223668771721570585118, 8.092912834085306122697436594710, 8.609504417396998023119856255437