| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 3·11-s − 3·13-s − 15-s + 3·17-s − 21-s + 23-s + 25-s + 5·27-s + 9·29-s + 2·31-s − 3·33-s + 35-s + 4·37-s + 3·39-s + 4·41-s + 2·43-s − 2·45-s + 13·47-s + 49-s − 3·51-s + 2·53-s + 3·55-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.832·13-s − 0.258·15-s + 0.727·17-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s + 0.359·31-s − 0.522·33-s + 0.169·35-s + 0.657·37-s + 0.480·39-s + 0.624·41-s + 0.304·43-s − 0.298·45-s + 1.89·47-s + 1/7·49-s − 0.420·51-s + 0.274·53-s + 0.404·55-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.669064695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669064695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.44041912190354, −14.19147192115801, −13.55599424599438, −12.83935494792298, −12.28113881402940, −11.95764558304751, −11.45640693234486, −11.00191052251672, −10.22005916173901, −10.01521625306642, −9.321732654418923, −8.610969600203587, −8.447570009843260, −7.440445639328562, −7.137483633226479, −6.353694590938309, −5.902347907505197, −5.464660566601361, −4.718025476606102, −4.356437242292180, −3.442691280735218, −2.697112501464693, −2.225002416506506, −1.114858728788691, −0.7018454719089312,
0.7018454719089312, 1.114858728788691, 2.225002416506506, 2.697112501464693, 3.442691280735218, 4.356437242292180, 4.718025476606102, 5.464660566601361, 5.902347907505197, 6.353694590938309, 7.137483633226479, 7.440445639328562, 8.447570009843260, 8.610969600203587, 9.321732654418923, 10.01521625306642, 10.22005916173901, 11.00191052251672, 11.45640693234486, 11.95764558304751, 12.28113881402940, 12.83935494792298, 13.55599424599438, 14.19147192115801, 14.44041912190354
