| L(s) = 1 | + 1.72·3-s + 0.586·5-s + 2.46·7-s − 0.0259·9-s − 2.73·11-s + 1.16·13-s + 1.01·15-s − 3.49·17-s − 5.19·19-s + 4.24·21-s + 1.89·23-s − 4.65·25-s − 5.21·27-s − 3.57·29-s + 1.26·31-s − 4.70·33-s + 1.44·35-s + 10.5·37-s + 2.00·39-s − 4.19·41-s − 8.57·43-s − 0.0152·45-s − 4.29·47-s − 0.944·49-s − 6.02·51-s − 6.91·53-s − 1.60·55-s + ⋯ |
| L(s) = 1 | + 0.995·3-s + 0.262·5-s + 0.930·7-s − 0.00864·9-s − 0.823·11-s + 0.322·13-s + 0.261·15-s − 0.847·17-s − 1.19·19-s + 0.926·21-s + 0.395·23-s − 0.931·25-s − 1.00·27-s − 0.663·29-s + 0.226·31-s − 0.819·33-s + 0.243·35-s + 1.73·37-s + 0.321·39-s − 0.654·41-s − 1.30·43-s − 0.00226·45-s − 0.626·47-s − 0.134·49-s − 0.843·51-s − 0.949·53-s − 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 0.586T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 + 8.57T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.80423649762654864631379822950, −6.86819450909751376193711337125, −6.10680975867818024054844832011, −5.34046125467184399400983144084, −4.58495749687462894153920863215, −3.89333281101821933957842380270, −2.94150766501646533810362417266, −2.23310393507246221682490563436, −1.63554498916254410383738748312, 0,
1.63554498916254410383738748312, 2.23310393507246221682490563436, 2.94150766501646533810362417266, 3.89333281101821933957842380270, 4.58495749687462894153920863215, 5.34046125467184399400983144084, 6.10680975867818024054844832011, 6.86819450909751376193711337125, 7.80423649762654864631379822950
