| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 2·11-s + 4·13-s − 14-s + 16-s − 6·17-s − 4·19-s − 20-s − 2·22-s + 23-s + 25-s + 4·26-s − 28-s − 2·29-s + 2·31-s + 32-s − 6·34-s + 35-s + 8·37-s − 4·38-s − 40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.169·35-s + 1.31·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372577902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372577902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.95863524013746, −15.48578884244769, −15.05265186777844, −14.56261690240726, −13.58297351938981, −13.28412261945855, −12.98898727000705, −12.24229407365333, −11.53694573247910, −11.09777842646009, −10.63381444493842, −9.963662293590851, −9.113700156444976, −8.492997893368319, −8.051730935893210, −7.169503216354242, −6.573592271037630, −6.151627074256641, −5.344768859748118, −4.580112021245800, −4.055146131140121, −3.378752405604127, −2.581984143487020, −1.851647438198656, −0.5842931609346309,
0.5842931609346309, 1.851647438198656, 2.581984143487020, 3.378752405604127, 4.055146131140121, 4.580112021245800, 5.344768859748118, 6.151627074256641, 6.573592271037630, 7.169503216354242, 8.051730935893210, 8.492997893368319, 9.113700156444976, 9.963662293590851, 10.63381444493842, 11.09777842646009, 11.53694573247910, 12.24229407365333, 12.98898727000705, 13.28412261945855, 13.58297351938981, 14.56261690240726, 15.05265186777844, 15.48578884244769, 15.95863524013746
