| L(s) = 1 | + 1.69·3-s + 2.39·5-s + 2.42·7-s − 0.110·9-s − 6.20·11-s − 2.05·13-s + 4.06·15-s + 3.12·17-s − 8.56·19-s + 4.12·21-s − 5.19·23-s + 0.724·25-s − 5.28·27-s − 0.325·29-s + 1.08·31-s − 10.5·33-s + 5.80·35-s + 6.84·37-s − 3.48·39-s − 4.00·41-s − 2.67·43-s − 0.265·45-s − 5.81·47-s − 1.10·49-s + 5.31·51-s + 10.8·53-s − 14.8·55-s + ⋯ |
| L(s) = 1 | + 0.981·3-s + 1.06·5-s + 0.917·7-s − 0.0369·9-s − 1.87·11-s − 0.569·13-s + 1.04·15-s + 0.758·17-s − 1.96·19-s + 0.900·21-s − 1.08·23-s + 0.144·25-s − 1.01·27-s − 0.0604·29-s + 0.194·31-s − 1.83·33-s + 0.981·35-s + 1.12·37-s − 0.558·39-s − 0.625·41-s − 0.408·43-s − 0.0395·45-s − 0.848·47-s − 0.158·49-s + 0.744·51-s + 1.49·53-s − 2.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 8.56T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 0.325T + 29T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 - 6.84T + 37T^{2} \) |
| 41 | \( 1 + 4.00T + 41T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.55T + 89T^{2} \) |
| 97 | \( 1 + 9.70T + 97T^{2} \) |
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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
−7.79931595674892555599329558104, −6.90203134143055408619154745117, −5.88481097243634283718287693762, −5.47461222307250069270109640314, −4.72318736468277811508216376250, −3.87446479927132520468606751018, −2.69126762231335343302444127044, −2.35564748839529428234826341021, −1.71223411844296335177745913467, 0,
1.71223411844296335177745913467, 2.35564748839529428234826341021, 2.69126762231335343302444127044, 3.87446479927132520468606751018, 4.72318736468277811508216376250, 5.47461222307250069270109640314, 5.88481097243634283718287693762, 6.90203134143055408619154745117, 7.79931595674892555599329558104
