L(s) = 1 | + 2-s + 0.162·3-s + 4-s + 2.16·5-s + 0.162·6-s − 0.548·7-s + 8-s − 2.97·9-s + 2.16·10-s + 1.31·11-s + 0.162·12-s − 0.548·14-s + 0.351·15-s + 16-s − 6.35·17-s − 2.97·18-s + 19-s + 2.16·20-s − 0.0892·21-s + 1.31·22-s − 7.53·23-s + 0.162·24-s − 0.318·25-s − 0.971·27-s − 0.548·28-s − 6.85·29-s + 0.351·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.0938·3-s + 0.5·4-s + 0.967·5-s + 0.0663·6-s − 0.207·7-s + 0.353·8-s − 0.991·9-s + 0.684·10-s + 0.395·11-s + 0.0469·12-s − 0.146·14-s + 0.0908·15-s + 0.250·16-s − 1.54·17-s − 0.700·18-s + 0.229·19-s + 0.483·20-s − 0.0194·21-s + 0.279·22-s − 1.57·23-s + 0.0331·24-s − 0.0636·25-s − 0.186·27-s − 0.103·28-s − 1.27·29-s + 0.0642·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.162T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 + 0.548T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 17 | \( 1 + 6.35T + 17T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 3.40T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 + 0.300T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 0.567T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 - 7.29T + 97T^{2} \) |
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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
−7.56001063513059187522964262528, −6.63341360422466477777929864610, −6.20895441080471182351837106031, −5.60914218659744992862147116754, −4.87557225738138589569542373642, −3.99046175039835504397116356726, −3.22486094089304179848934175237, −2.27087576721479829322886670426, −1.76637526984031914021543557587, 0,
1.76637526984031914021543557587, 2.27087576721479829322886670426, 3.22486094089304179848934175237, 3.99046175039835504397116356726, 4.87557225738138589569542373642, 5.60914218659744992862147116754, 6.20895441080471182351837106031, 6.63341360422466477777929864610, 7.56001063513059187522964262528