| L(s) = 1 | − 7-s − 3·9-s + 6·13-s + 4·17-s − 4·19-s + 4·23-s + 2·31-s − 2·37-s − 10·41-s + 8·43-s + 49-s − 2·53-s − 14·61-s + 3·63-s + 2·67-s + 4·71-s − 4·73-s + 14·79-s + 9·81-s + 18·83-s + 14·89-s − 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.66·13-s + 0.970·17-s − 0.917·19-s + 0.834·23-s + 0.359·31-s − 0.328·37-s − 1.56·41-s + 1.21·43-s + 1/7·49-s − 0.274·53-s − 1.79·61-s + 0.377·63-s + 0.244·67-s + 0.474·71-s − 0.468·73-s + 1.57·79-s + 81-s + 1.97·83-s + 1.48·89-s − 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.608069191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.608069191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.49267312437872, −12.10340064751960, −11.75003490585131, −11.09063624628256, −10.71133796684127, −10.57023015922770, −9.834703159072375, −9.244781354680781, −8.892413726872517, −8.554118816823430, −7.969660800121014, −7.686593094436419, −6.872001734755106, −6.420442196086425, −6.086605212590049, −5.664208989557312, −5.049629550397996, −4.593666979708540, −3.803109041968226, −3.364396298200004, −3.140709019857485, −2.326838849815143, −1.752417018340810, −0.9911230929752992, −0.4880792080200617,
0.4880792080200617, 0.9911230929752992, 1.752417018340810, 2.326838849815143, 3.140709019857485, 3.364396298200004, 3.803109041968226, 4.593666979708540, 5.049629550397996, 5.664208989557312, 6.086605212590049, 6.420442196086425, 6.872001734755106, 7.686593094436419, 7.969660800121014, 8.554118816823430, 8.892413726872517, 9.244781354680781, 9.834703159072375, 10.57023015922770, 10.71133796684127, 11.09063624628256, 11.75003490585131, 12.10340064751960, 12.49267312437872
