| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 3.25·5-s − 1.41·6-s − 8-s − 0.999·9-s + 3.25·10-s + 4.60·11-s + 1.41·12-s − 3.25·13-s − 4.60·15-s + 16-s + 0.986·17-s + 0.999·18-s + 4.24·19-s − 3.25·20-s − 4.60·22-s + 2·23-s − 1.41·24-s + 5.60·25-s + 3.25·26-s − 5.65·27-s + 29-s + 4.60·30-s − 6.94·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s − 1.45·5-s − 0.577·6-s − 0.353·8-s − 0.333·9-s + 1.02·10-s + 1.38·11-s + 0.408·12-s − 0.903·13-s − 1.18·15-s + 0.250·16-s + 0.239·17-s + 0.235·18-s + 0.973·19-s − 0.728·20-s − 0.981·22-s + 0.417·23-s − 0.288·24-s + 1.12·25-s + 0.638·26-s − 1.08·27-s + 0.185·29-s + 0.840·30-s − 1.24·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.162976113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162976113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 0.986T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 31 | \( 1 + 6.94T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 0.986T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 4.67T + 59T^{2} \) |
| 61 | \( 1 + 7.36T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 + 8.35T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 8.35T + 97T^{2} \) |
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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
−8.821560506381785697758121395584, −8.066552931657223895703449454568, −7.39947008142599412802258580653, −7.02617749708690903150362536174, −5.83317601787910221743867248504, −4.71925633347122719527407728118, −3.65653565923889689597562804356, −3.26802642536924494382102264845, −2.06941214330554972948628902682, −0.70217374752943321891385308583,
0.70217374752943321891385308583, 2.06941214330554972948628902682, 3.26802642536924494382102264845, 3.65653565923889689597562804356, 4.71925633347122719527407728118, 5.83317601787910221743867248504, 7.02617749708690903150362536174, 7.39947008142599412802258580653, 8.066552931657223895703449454568, 8.821560506381785697758121395584
