L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 2·13-s − 6·17-s − 8·19-s − 2·21-s − 4·23-s + 27-s + 8·29-s − 2·33-s + 10·37-s − 2·39-s + 2·41-s − 12·43-s − 3·49-s − 6·51-s − 10·53-s − 8·57-s + 6·59-s + 2·61-s − 2·63-s − 8·67-s − 4·69-s + 4·71-s − 4·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s + 0.192·27-s + 1.48·29-s − 0.348·33-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s − 3/7·49-s − 0.840·51-s − 1.37·53-s − 1.05·57-s + 0.781·59-s + 0.256·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s + 0.474·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.343061317538849855270026145596, −8.473810258862941042294031254575, −7.900639168738143112286151233174, −6.66208513585632291920486678528, −6.31249851341993462828303401256, −4.82757705586149563119321113004, −4.12182189313281906026607987035, −2.86526191126796724123303244993, −2.11268904766511061908179462251, 0,
2.11268904766511061908179462251, 2.86526191126796724123303244993, 4.12182189313281906026607987035, 4.82757705586149563119321113004, 6.31249851341993462828303401256, 6.66208513585632291920486678528, 7.900639168738143112286151233174, 8.473810258862941042294031254575, 9.343061317538849855270026145596