| L(s) = 1 | + 0.442·2-s − 2.59·3-s − 1.80·4-s + 5-s − 1.14·6-s + 1.77·7-s − 1.68·8-s + 3.75·9-s + 0.442·10-s − 5.16·11-s + 4.68·12-s + 3.86·13-s + 0.784·14-s − 2.59·15-s + 2.86·16-s + 5.69·17-s + 1.66·18-s − 1.54·19-s − 1.80·20-s − 4.60·21-s − 2.28·22-s − 6.38·23-s + 4.37·24-s + 25-s + 1.70·26-s − 1.96·27-s − 3.19·28-s + ⋯ |
| L(s) = 1 | + 0.312·2-s − 1.50·3-s − 0.902·4-s + 0.447·5-s − 0.469·6-s + 0.669·7-s − 0.594·8-s + 1.25·9-s + 0.139·10-s − 1.55·11-s + 1.35·12-s + 1.07·13-s + 0.209·14-s − 0.671·15-s + 0.716·16-s + 1.38·17-s + 0.391·18-s − 0.354·19-s − 0.403·20-s − 1.00·21-s − 0.487·22-s − 1.33·23-s + 0.892·24-s + 0.200·25-s + 0.334·26-s − 0.378·27-s − 0.604·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.442T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 7 | \( 1 - 1.77T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 0.248T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 1.78T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 + 1.42T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−9.574508947370608478245320632242, −8.323039833560312837598065603395, −7.86097273909009446502003643988, −6.39890825394739441136000400035, −5.72328977259048526054727892941, −5.19193932083744894294336957593, −4.51167722017905409037846633962, −3.24506338538497644304375216001, −1.44593174098119538028348729136, 0,
1.44593174098119538028348729136, 3.24506338538497644304375216001, 4.51167722017905409037846633962, 5.19193932083744894294336957593, 5.72328977259048526054727892941, 6.39890825394739441136000400035, 7.86097273909009446502003643988, 8.323039833560312837598065603395, 9.574508947370608478245320632242
