L(s) = 1 | + 2-s + 0.214·3-s + 4-s − 5-s + 0.214·6-s − 0.767·7-s + 8-s − 2.95·9-s − 10-s − 2.29·11-s + 0.214·12-s + 1.53·13-s − 0.767·14-s − 0.214·15-s + 16-s + 6.58·17-s − 2.95·18-s + 0.141·19-s − 20-s − 0.164·21-s − 2.29·22-s + 4.09·23-s + 0.214·24-s + 25-s + 1.53·26-s − 1.27·27-s − 0.767·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.123·3-s + 0.5·4-s − 0.447·5-s + 0.0874·6-s − 0.290·7-s + 0.353·8-s − 0.984·9-s − 0.316·10-s − 0.693·11-s + 0.0618·12-s + 0.425·13-s − 0.205·14-s − 0.0552·15-s + 0.250·16-s + 1.59·17-s − 0.696·18-s + 0.0323·19-s − 0.223·20-s − 0.0358·21-s − 0.490·22-s + 0.853·23-s + 0.0437·24-s + 0.200·25-s + 0.300·26-s − 0.245·27-s − 0.145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 0.214T + 3T^{2} \) |
| 7 | \( 1 + 0.767T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 - 0.141T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 6.75T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 0.700T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 - 2.70T + 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 - 1.87T + 89T^{2} \) |
| 97 | \( 1 + 1.02T + 97T^{2} \) |
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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
−7.66536912344977263875023475272, −7.07543772615063318366554571971, −6.09750378448073676840937016958, −5.48584869039618857752190970793, −5.02690324759772109388925112230, −3.81215529438066503693109177281, −3.32836467151623499943324508727, −2.66511282778498999783703206080, −1.44881063407764746974469405670, 0,
1.44881063407764746974469405670, 2.66511282778498999783703206080, 3.32836467151623499943324508727, 3.81215529438066503693109177281, 5.02690324759772109388925112230, 5.48584869039618857752190970793, 6.09750378448073676840937016958, 7.07543772615063318366554571971, 7.66536912344977263875023475272