| L(s) = 1 | − 5-s + 5·11-s + 2·13-s + 6·17-s + 2·19-s − 6·23-s − 4·25-s − 3·29-s + 5·31-s − 2·37-s + 8·41-s − 4·43-s + 4·47-s + 9·53-s − 5·55-s − 3·59-s − 12·61-s − 2·65-s + 2·67-s − 8·71-s − 14·73-s + 79-s + 17·83-s − 6·85-s + 18·89-s − 2·95-s + 3·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s − 4/5·25-s − 0.557·29-s + 0.898·31-s − 0.328·37-s + 1.24·41-s − 0.609·43-s + 0.583·47-s + 1.23·53-s − 0.674·55-s − 0.390·59-s − 1.53·61-s − 0.248·65-s + 0.244·67-s − 0.949·71-s − 1.63·73-s + 0.112·79-s + 1.86·83-s − 0.650·85-s + 1.90·89-s − 0.205·95-s + 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.006197179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006197179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.555818845397792556065203605186, −7.76097452456710202642080255842, −7.25783162427823859188080698277, −6.11426655564976949143677627155, −5.85528932752195064495220697038, −4.58173483434378937823868417435, −3.83405358561613399740750865388, −3.27774564304333526952479821216, −1.86264503136815553981124561216, −0.881389698673377256974954119311,
0.881389698673377256974954119311, 1.86264503136815553981124561216, 3.27774564304333526952479821216, 3.83405358561613399740750865388, 4.58173483434378937823868417435, 5.85528932752195064495220697038, 6.11426655564976949143677627155, 7.25783162427823859188080698277, 7.76097452456710202642080255842, 8.555818845397792556065203605186
