| L(s) = 1 | + 5-s − 7-s − 3·9-s − 4·11-s + 6·13-s + 2·17-s + 25-s − 6·29-s + 8·31-s − 35-s + 10·37-s + 2·41-s − 4·43-s − 3·45-s + 8·47-s + 49-s + 2·53-s − 4·55-s + 8·59-s + 14·61-s + 3·63-s + 6·65-s + 12·67-s − 16·71-s + 2·73-s + 4·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s − 1.20·11-s + 1.66·13-s + 0.485·17-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.169·35-s + 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s + 1.04·59-s + 1.79·61-s + 0.377·63-s + 0.744·65-s + 1.46·67-s − 1.89·71-s + 0.234·73-s + 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669202414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669202414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.910824986105461683565850952843, −8.342925504546681427970604924895, −7.63810043151096799752774274091, −6.50661147582867110066747443111, −5.82344169953684002039972565231, −5.35297783463712406492518278139, −4.07505442753619333270812979660, −3.11732069850814337090071774581, −2.34423344984301921891298897662, −0.837608542604674414323310984226,
0.837608542604674414323310984226, 2.34423344984301921891298897662, 3.11732069850814337090071774581, 4.07505442753619333270812979660, 5.35297783463712406492518278139, 5.82344169953684002039972565231, 6.50661147582867110066747443111, 7.63810043151096799752774274091, 8.342925504546681427970604924895, 8.910824986105461683565850952843
