| L(s) = 1 | − 1.73·3-s − 11-s − 1.73·13-s + 5.19·17-s + 2.82·19-s − 2.44·23-s + 5.19·27-s − 7·29-s − 7.07·31-s + 1.73·33-s + 7.34·37-s + 2.99·39-s − 7.07·41-s + 9.79·43-s + 12.1·47-s − 9·51-s − 12.2·53-s − 4.89·57-s + 7.07·59-s + 14.1·61-s − 12.2·67-s + 4.24·69-s − 10·71-s + 3·79-s − 9·81-s + 12.1·87-s + 12.2·93-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 0.301·11-s − 0.480·13-s + 1.26·17-s + 0.648·19-s − 0.510·23-s + 1.00·27-s − 1.29·29-s − 1.27·31-s + 0.301·33-s + 1.20·37-s + 0.480·39-s − 1.10·41-s + 1.49·43-s + 1.76·47-s − 1.26·51-s − 1.68·53-s − 0.648·57-s + 0.920·59-s + 1.81·61-s − 1.49·67-s + 0.510·69-s − 1.18·71-s + 0.337·79-s − 81-s + 1.29·87-s + 1.27·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−7.58903664122413141252899155439, −7.37193220635309285187918684713, −6.24958169386000890088192215401, −5.57722798947776012493709630609, −5.29565266391275389897489623987, −4.24672238844000599419842989750, −3.36155928812820878832722646778, −2.38462877806322202544389278952, −1.14577402262443185568968011369, 0,
1.14577402262443185568968011369, 2.38462877806322202544389278952, 3.36155928812820878832722646778, 4.24672238844000599419842989750, 5.29565266391275389897489623987, 5.57722798947776012493709630609, 6.24958169386000890088192215401, 7.37193220635309285187918684713, 7.58903664122413141252899155439
