L(s) = 1 | − 0.921·2-s − 3-s − 1.15·4-s + 0.921·6-s − 3.92·7-s + 2.90·8-s + 9-s + 5.05·11-s + 1.15·12-s + 0.610·13-s + 3.61·14-s − 0.371·16-s + 2.84·17-s − 0.921·18-s − 3.27·19-s + 3.92·21-s − 4.65·22-s − 4.90·23-s − 2.90·24-s − 0.562·26-s − 27-s + 4.51·28-s − 4.76·29-s + 3.72·31-s − 5.46·32-s − 5.05·33-s − 2.62·34-s + ⋯ |
L(s) = 1 | − 0.651·2-s − 0.577·3-s − 0.575·4-s + 0.376·6-s − 1.48·7-s + 1.02·8-s + 0.333·9-s + 1.52·11-s + 0.332·12-s + 0.169·13-s + 0.965·14-s − 0.0928·16-s + 0.690·17-s − 0.217·18-s − 0.750·19-s + 0.855·21-s − 0.993·22-s − 1.02·23-s − 0.592·24-s − 0.110·26-s − 0.192·27-s + 0.853·28-s − 0.885·29-s + 0.669·31-s − 0.965·32-s − 0.880·33-s − 0.449·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
good | 2 | \( 1 + 0.921T + 2T^{2} \) |
| 7 | \( 1 + 3.92T + 7T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 0.610T + 13T^{2} \) |
| 17 | \( 1 - 2.84T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 - 3.72T + 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 - 0.562T + 41T^{2} \) |
| 43 | \( 1 - 2.21T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 + 8.67T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 - 8.41T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 + 0.141T + 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−7.44139655130600616248985543166, −6.76774468456729094848036695745, −6.17133040442170845862038759469, −5.63092686590551125536103293942, −4.51154643496623855448356288953, −3.91868709943778826702464899029, −3.32922292936674818570242607099, −1.92046120617425424863477604354, −0.941275568110180687463574683242, 0,
0.941275568110180687463574683242, 1.92046120617425424863477604354, 3.32922292936674818570242607099, 3.91868709943778826702464899029, 4.51154643496623855448356288953, 5.63092686590551125536103293942, 6.17133040442170845862038759469, 6.76774468456729094848036695745, 7.44139655130600616248985543166