| L(s) = 1 | − 0.732·2-s − 3-s − 1.46·4-s + 0.732·6-s + 2.53·8-s + 9-s − 2.73·11-s + 1.46·12-s + 5.73·13-s + 1.07·16-s + 6.73·17-s − 0.732·18-s + 2.46·19-s + 2·22-s + 1.26·23-s − 2.53·24-s − 4.19·26-s − 27-s + 6.19·29-s − 6.46·31-s − 5.85·32-s + 2.73·33-s − 4.92·34-s − 1.46·36-s − 7.19·37-s − 1.80·38-s − 5.73·39-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.577·3-s − 0.732·4-s + 0.298·6-s + 0.896·8-s + 0.333·9-s − 0.823·11-s + 0.422·12-s + 1.58·13-s + 0.267·16-s + 1.63·17-s − 0.172·18-s + 0.565·19-s + 0.426·22-s + 0.264·23-s − 0.517·24-s − 0.822·26-s − 0.192·27-s + 1.15·29-s − 1.16·31-s − 1.03·32-s + 0.475·33-s − 0.845·34-s − 0.244·36-s − 1.18·37-s − 0.292·38-s − 0.917·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013215265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013215265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 7.19T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−8.634600473453895480720660911993, −7.78346763124498357013353427450, −7.33301485803695745849939701754, −6.14920323733806161412877523239, −5.50603161033232513432509219587, −4.91455032616381280109438159151, −3.86775738177741210899360661204, −3.17072792592381130922122229488, −1.53609626877493731914642008821, −0.72311973202418744922349492473,
0.72311973202418744922349492473, 1.53609626877493731914642008821, 3.17072792592381130922122229488, 3.86775738177741210899360661204, 4.91455032616381280109438159151, 5.50603161033232513432509219587, 6.14920323733806161412877523239, 7.33301485803695745849939701754, 7.78346763124498357013353427450, 8.634600473453895480720660911993
