| L(s) = 1 | + 2.53·2-s + 4.43·4-s − 1.04·7-s + 6.19·8-s − 2.97·11-s − 5.66·13-s − 2.64·14-s + 6.83·16-s − 5.08·17-s − 5.37·19-s − 7.53·22-s + 3.86·23-s − 14.3·26-s − 4.61·28-s − 0.679·29-s + 0.850·31-s + 4.95·32-s − 12.9·34-s + 1.61·37-s − 13.6·38-s − 1.16·41-s − 5.68·43-s − 13.1·44-s + 9.79·46-s + 3.28·47-s − 5.91·49-s − 25.1·52-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.21·4-s − 0.393·7-s + 2.18·8-s − 0.895·11-s − 1.57·13-s − 0.705·14-s + 1.70·16-s − 1.23·17-s − 1.23·19-s − 1.60·22-s + 0.804·23-s − 2.82·26-s − 0.873·28-s − 0.126·29-s + 0.152·31-s + 0.875·32-s − 2.21·34-s + 0.265·37-s − 2.21·38-s − 0.181·41-s − 0.867·43-s − 1.98·44-s + 1.44·46-s + 0.479·47-s − 0.845·49-s − 3.49·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 - 3.86T + 23T^{2} \) |
| 29 | \( 1 + 0.679T + 29T^{2} \) |
| 31 | \( 1 - 0.850T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 - 0.929T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.44T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 - 6.02T + 97T^{2} \) |
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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
−7.29499568912635479254249514098, −6.95697084561280714720713265300, −6.18475801123451740826103600324, −5.46766561742165304390889888961, −4.67727735508446125160082051438, −4.40672719429814087819152226148, −3.27796150745624589964063510393, −2.57612565900687711245122363190, −2.03431065185049761826468809371, 0,
2.03431065185049761826468809371, 2.57612565900687711245122363190, 3.27796150745624589964063510393, 4.40672719429814087819152226148, 4.67727735508446125160082051438, 5.46766561742165304390889888961, 6.18475801123451740826103600324, 6.95697084561280714720713265300, 7.29499568912635479254249514098
