L(s) = 1 | − 1.92·2-s − 0.860·3-s + 1.72·4-s + 1.66·6-s + 4.27·7-s + 0.532·8-s − 2.25·9-s − 4.15·11-s − 1.48·12-s − 1.01·13-s − 8.25·14-s − 4.47·16-s − 5.08·17-s + 4.35·18-s − 0.226·19-s − 3.68·21-s + 8.02·22-s + 2.25·23-s − 0.458·24-s + 1.95·26-s + 4.52·27-s + 7.37·28-s − 0.0980·29-s + 5.03·31-s + 7.57·32-s + 3.58·33-s + 9.82·34-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 0.497·3-s + 0.861·4-s + 0.678·6-s + 1.61·7-s + 0.188·8-s − 0.752·9-s − 1.25·11-s − 0.428·12-s − 0.281·13-s − 2.20·14-s − 1.11·16-s − 1.23·17-s + 1.02·18-s − 0.0520·19-s − 0.803·21-s + 1.71·22-s + 0.470·23-s − 0.0936·24-s + 0.383·26-s + 0.871·27-s + 1.39·28-s − 0.0182·29-s + 0.905·31-s + 1.33·32-s + 0.623·33-s + 1.68·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 + 0.860T + 3T^{2} \) |
| 7 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 0.226T + 19T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 + 0.0980T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 + 0.598T + 83T^{2} \) |
| 89 | \( 1 - 4.03T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−7.86428233335964302152816206642, −7.38235326076402630100355884943, −6.48128269817396910593663978679, −5.54975338966373957727888146654, −4.85119311591595047095230901794, −4.39911840414514454046899715130, −2.71381043846811850640646854312, −2.12457994108672616023204691472, −1.03154200187763773647899555819, 0,
1.03154200187763773647899555819, 2.12457994108672616023204691472, 2.71381043846811850640646854312, 4.39911840414514454046899715130, 4.85119311591595047095230901794, 5.54975338966373957727888146654, 6.48128269817396910593663978679, 7.38235326076402630100355884943, 7.86428233335964302152816206642