| L(s) = 1 | + 3-s + 9-s − 2·11-s + 13-s + 17-s + 8·19-s + 6·23-s + 27-s + 4·31-s − 2·33-s − 2·37-s + 39-s + 4·41-s − 4·43-s − 7·49-s + 51-s + 6·53-s + 8·57-s − 4·59-s + 8·61-s + 8·67-s + 6·69-s + 4·71-s − 10·79-s + 81-s + 8·83-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 1.25·23-s + 0.192·27-s + 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s − 49-s + 0.140·51-s + 0.824·53-s + 1.05·57-s − 0.520·59-s + 1.02·61-s + 0.977·67-s + 0.722·69-s + 0.474·71-s − 1.12·79-s + 1/9·81-s + 0.878·83-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.451620284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.451620284\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.98486704880128, −12.37388100398436, −11.77115884397701, −11.58190419084046, −10.75010529008576, −10.66354232057492, −9.807992659629870, −9.599686966821062, −9.216074839604655, −8.373757553474622, −8.304625012407825, −7.678500325260039, −7.107026342189796, −6.910014822193324, −6.145809711413016, −5.520419620424264, −5.161601036310359, −4.678452573230067, −4.004889014384915, −3.310465244120490, −3.092751956547737, −2.505550170627492, −1.788811037836586, −1.096810429823017, −0.6141093393537403,
0.6141093393537403, 1.096810429823017, 1.788811037836586, 2.505550170627492, 3.092751956547737, 3.310465244120490, 4.004889014384915, 4.678452573230067, 5.161601036310359, 5.520419620424264, 6.145809711413016, 6.910014822193324, 7.107026342189796, 7.678500325260039, 8.304625012407825, 8.373757553474622, 9.216074839604655, 9.599686966821062, 9.807992659629870, 10.66354232057492, 10.75010529008576, 11.58190419084046, 11.77115884397701, 12.37388100398436, 12.98486704880128
