L(s) = 1 | − 2·3-s + 2·7-s + 9-s − 2·13-s + 6·17-s + 4·19-s − 4·21-s + 6·23-s + 4·27-s + 6·29-s + 4·31-s − 2·37-s + 4·39-s + 6·41-s − 10·43-s − 6·47-s − 3·49-s − 12·51-s + 6·53-s − 8·57-s − 12·59-s + 2·61-s + 2·63-s + 2·67-s − 12·69-s + 12·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.872·21-s + 1.25·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 1.52·43-s − 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-s + 0.244·67-s − 1.44·69-s + 1.42·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.017037595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017037595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−11.43602784825647007842815601932, −10.49081175144606776577808780778, −9.703656252458157439228300963800, −8.407676785947288187996164912905, −7.47121424818625422471458115292, −6.43307814142801241380910059979, −5.30602566274475700528497955431, −4.82118726362395324834121081499, −3.07875775618643284846845606551, −1.10363850080882238424508077779,
1.10363850080882238424508077779, 3.07875775618643284846845606551, 4.82118726362395324834121081499, 5.30602566274475700528497955431, 6.43307814142801241380910059979, 7.47121424818625422471458115292, 8.407676785947288187996164912905, 9.703656252458157439228300963800, 10.49081175144606776577808780778, 11.43602784825647007842815601932