| L(s) = 1 | + 0.841·5-s − 1.64·7-s + 4.75·11-s − 3.74·13-s + 5.53·17-s − 5.15·19-s + 4.33·23-s − 4.29·25-s + 8.66·29-s + 5.64·31-s − 1.38·35-s + 0.913·37-s − 7.91·41-s + 0.500·43-s + 9.10·47-s − 4.29·49-s + 6.97·53-s + 3.99·55-s − 6.13·59-s − 0.913·61-s − 3.14·65-s − 5.65·67-s − 13.4·71-s − 3.29·73-s − 7.82·77-s + 9.64·79-s + 4.75·83-s + ⋯ |
| L(s) = 1 | + 0.376·5-s − 0.622·7-s + 1.43·11-s − 1.03·13-s + 1.34·17-s − 1.18·19-s + 0.904·23-s − 0.858·25-s + 1.60·29-s + 1.01·31-s − 0.234·35-s + 0.150·37-s − 1.23·41-s + 0.0763·43-s + 1.32·47-s − 0.613·49-s + 0.958·53-s + 0.539·55-s − 0.799·59-s − 0.116·61-s − 0.390·65-s − 0.691·67-s − 1.59·71-s − 0.385·73-s − 0.891·77-s + 1.08·79-s + 0.521·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.188243751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188243751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.841T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 - 4.33T + 23T^{2} \) |
| 29 | \( 1 - 8.66T + 29T^{2} \) |
| 31 | \( 1 - 5.64T + 31T^{2} \) |
| 37 | \( 1 - 0.913T + 37T^{2} \) |
| 41 | \( 1 + 7.91T + 41T^{2} \) |
| 43 | \( 1 - 0.500T + 43T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 61 | \( 1 + 0.913T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 2.38T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.62492671045799516850383195636, −6.93331170105454839518951917769, −6.35646577615019685518758391888, −5.84867641021576918057082319424, −4.86704066545915544156188652467, −4.27342129287598868978797684578, −3.37143004792670691943727050167, −2.69971533691980603114463741435, −1.69793796335124873103427171773, −0.72910350531520943437754946736,
0.72910350531520943437754946736, 1.69793796335124873103427171773, 2.69971533691980603114463741435, 3.37143004792670691943727050167, 4.27342129287598868978797684578, 4.86704066545915544156188652467, 5.84867641021576918057082319424, 6.35646577615019685518758391888, 6.93331170105454839518951917769, 7.62492671045799516850383195636
