| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 6·11-s − 13-s + 15-s + 6·17-s + 7·19-s − 21-s + 6·23-s + 25-s − 27-s + 9·29-s + 4·31-s + 6·33-s − 35-s + 39-s + 6·41-s − 2·43-s − 45-s − 12·47-s − 6·49-s − 6·51-s + 6·53-s + 6·55-s − 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s + 1.60·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 0.718·31-s + 1.04·33-s − 0.169·35-s + 0.160·39-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 1.75·47-s − 6/7·49-s − 0.840·51-s + 0.824·53-s + 0.809·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.395666434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.395666434\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.71870176228326, −11.92763317527213, −11.78550548060929, −11.31769497064197, −10.82437746168731, −10.26672533158223, −9.949481057944224, −9.705235247503329, −8.862037681206685, −8.204075638482428, −8.052874218931220, −7.534990654023755, −7.112243476453565, −6.648383456266882, −5.902903863584414, −5.367657498165610, −5.068473411350331, −4.830597296404633, −4.092040506213331, −3.280494197128831, −2.956321419537035, −2.547517458283505, −1.552272239499749, −0.9247408156769309, −0.5369315513941840,
0.5369315513941840, 0.9247408156769309, 1.552272239499749, 2.547517458283505, 2.956321419537035, 3.280494197128831, 4.092040506213331, 4.830597296404633, 5.068473411350331, 5.367657498165610, 5.902903863584414, 6.648383456266882, 7.112243476453565, 7.534990654023755, 8.052874218931220, 8.204075638482428, 8.862037681206685, 9.705235247503329, 9.949481057944224, 10.26672533158223, 10.82437746168731, 11.31769497064197, 11.78550548060929, 11.92763317527213, 12.71870176228326
