| L(s) = 1 | − 1.81·2-s + 3-s + 1.28·4-s + 1.10·5-s − 1.81·6-s + 1.28·8-s + 9-s − 2·10-s + 0.813·11-s + 1.28·12-s − 5.10·13-s + 1.10·15-s − 4.91·16-s − 3.39·17-s − 1.81·18-s − 3.10·19-s + 1.42·20-s − 1.47·22-s − 0.897·23-s + 1.28·24-s − 3.78·25-s + 9.25·26-s + 27-s + 4.44·29-s − 2·30-s + 8.96·31-s + 6.33·32-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.577·3-s + 0.644·4-s + 0.493·5-s − 0.740·6-s + 0.455·8-s + 0.333·9-s − 0.632·10-s + 0.245·11-s + 0.372·12-s − 1.41·13-s + 0.284·15-s − 1.22·16-s − 0.822·17-s − 0.427·18-s − 0.711·19-s + 0.317·20-s − 0.314·22-s − 0.187·23-s + 0.263·24-s − 0.756·25-s + 1.81·26-s + 0.192·27-s + 0.824·29-s − 0.365·30-s + 1.61·31-s + 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 - 0.813T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 + 0.897T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 0.235T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 + 1.91T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 7.68T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−7.87300271648913760931621048517, −7.24513233272592872275306932976, −6.62785510414224473596620669061, −5.76165040722357987498405445808, −4.55456438716255186101379185128, −4.26276476813171114000745602022, −2.68790432287775649397997189871, −2.27315507031838323711502170154, −1.25011270715050356204940204496, 0,
1.25011270715050356204940204496, 2.27315507031838323711502170154, 2.68790432287775649397997189871, 4.26276476813171114000745602022, 4.55456438716255186101379185128, 5.76165040722357987498405445808, 6.62785510414224473596620669061, 7.24513233272592872275306932976, 7.87300271648913760931621048517
