| L(s) = 1 | − 3-s − 2·7-s + 9-s + 11-s + 13-s + 6·17-s − 2·19-s + 2·21-s − 5·25-s − 27-s + 6·29-s + 10·31-s − 33-s + 2·37-s − 39-s − 8·43-s − 12·47-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s + 2·61-s − 2·63-s − 2·67-s − 10·73-s + 5·75-s − 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.174·33-s + 0.328·37-s − 0.160·39-s − 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 0.256·61-s − 0.251·63-s − 0.244·67-s − 1.17·73-s + 0.577·75-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422949079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422949079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.035452209053506985676129443229, −7.13730333332887580488158805783, −6.34959723647648537630514929537, −6.09773566072811756541764653366, −5.15007348650444140933557503089, −4.43320604464580760463866729393, −3.53988268503975191302056017866, −2.89166351410651270219620861787, −1.64044105248036400082215023240, −0.64750561277920242197015995024,
0.64750561277920242197015995024, 1.64044105248036400082215023240, 2.89166351410651270219620861787, 3.53988268503975191302056017866, 4.43320604464580760463866729393, 5.15007348650444140933557503089, 6.09773566072811756541764653366, 6.34959723647648537630514929537, 7.13730333332887580488158805783, 8.035452209053506985676129443229
