| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s − 2·9-s − 12-s + 13-s + 2·14-s + 16-s + 17-s + 2·18-s − 19-s + 2·21-s + 6·23-s + 24-s − 26-s + 5·27-s − 2·28-s − 3·29-s + 5·31-s − 32-s − 34-s − 2·36-s − 8·37-s + 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.436·21-s + 1.25·23-s + 0.204·24-s − 0.196·26-s + 0.962·27-s − 0.377·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.171·34-s − 1/3·36-s − 1.31·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7079651961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7079651961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−10.26666704357266149533944282295, −9.254655168764339561555644961985, −8.701731650384840424746561254406, −7.63448204361690073025409493735, −6.69860566748043802940529109511, −6.00511421282862527652019360354, −5.08559863510675804592773692866, −3.59551141535474323755203179860, −2.52827889076752744542650486558, −0.76105700396433611244376706065,
0.76105700396433611244376706065, 2.52827889076752744542650486558, 3.59551141535474323755203179860, 5.08559863510675804592773692866, 6.00511421282862527652019360354, 6.69860566748043802940529109511, 7.63448204361690073025409493735, 8.701731650384840424746561254406, 9.254655168764339561555644961985, 10.26666704357266149533944282295
